Interactive Manual › Chapter 13

Chapter 13: Inferential Statistics and Distributions

Hypothesis tests and confidence intervals (Z, T, proportion, chi-square, F, regression and ANOVA), their input editors and output variables, plus the DISTR probability functions and distribution shading.

Getting Started: Mean Height of a Populationp. 215

This quick tour estimates the average height of a population of women from a random sample. Heights in a biological population are roughly normally distributed, so a t confidence interval is the right tool for estimating the mean. You will also use the normal distribution to find a percentile and shade it on a graph.

The sample is the first ten of 90 values generated from a normal population with mean 165.1 cm and standard deviation 6.35 cm: 169.43, 168.33, 159.55, 169.97, 159.79, 181.42, 171.17, 162.04, 167.15 and 159.53.

Tutorial

Store the ten heights in a new list named HGHT

You will make a new list in the stat list editor and type the data into it.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the stat list editor.
STATENTER
Move the cursor up onto the L₁ column name, then press 2ndINS (INS) to insert a new list. A Name= prompt appears and alpha-lock is already on.
▲︎2ndINS
Type the name HGHT. Alpha-lock is on, so just press the keys whose green letters are H, G, H and T: ^ TAN ^ 4. Do not press ALPHA again.
^TAN^4
Press ENTER to create the list. The other lists shift one column to the right.
ENTER
Move down into the first row. The bottom line shows HGHT(1)=. Type the first height, 169.43, and press ENTER.
▼︎169.43ENTER
Enter the other nine heights the same way: 168.33, 159.55, 169.97, 159.79, 181.42, 171.17, 162.04, 167.15 and 159.53, each followed by ENTER.
168.33ENTER159.55ENTER169.97ENTER159.79ENTER181.42ENTER171.17ENTER162.04ENTER167.15ENTER159.53ENTER

Tutorial

A 99% t confidence interval from the list data

TInterval estimates a population mean when the standard deviation is unknown. Here it uses the data in HGHT.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the TInterval editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 8:TInterval by pressing 8.
STAT▶︎▶︎8
Inpt: should have Data highlighted (the default). If it does not, press ◀︎ ENTER. Then move down to List: and press 2ndLIST to open the list names menu; HGHT is item 7. Press 7.
▼︎2ndLIST7
Press ENTER to accept the list name. The cursor moves to Freq:, which stays at 1 (every value counted once).
ENTER
Move down to C-Level: and type .99 for a 99 percent confidence level. Press ▼︎ to reach Calculate.
▼︎.99▼︎
Press ENTER on Calculate. The first line is the interval, about (159.74, 173.94): a spread of roughly 14.2 cm. The lines below give the sample mean, the sample standard deviation Sx and the sample size n.
ENTER

The 0.99 confidence level means that in a very large number of samples, about 99 percent of the intervals calculated this way would contain the true population mean. The real mean here (165.1 cm) does lie inside the interval.

To pin the mean down more precisely, use a bigger sample. The full set of 90 values has mean 163.8 and standard deviation 7.1. Rather than typing 90 values you can feed those summary statistics straight in with the Stats input option.

Tutorial

The same interval from summary statistics (n = 90)

Use Inpt: Stats to supply x̄, Sx and n directly.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the TInterval editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 8:TInterval by pressing 8.
STAT▶︎▶︎8
Highlight Stats next to Inpt: with ▶︎ and press ENTER. The List and Freq rows are replaced by rows for the sample mean x̄, Sx and n.
▶︎ENTER
Move down and enter x̄ = 163.8, Sx = 7.1 and n = 90, pressing ENTER after each.
▼︎163.8ENTER7.1ENTER90ENTER
The cursor is on C-Level:. Enter .99 again, then press ENTER to move onto Calculate.
.99ENTER
Press ENTER to calculate. The interval is narrower than before, about (161.83, 165.77).
ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎8◀︎ENTER

Now a question about the whole population. If heights are normal with μ = 165.1 cm and σ = 6.35 cm, what height is exceeded by only 5 percent of women (the 95th percentile)?

Tutorial

Find the 95th percentile with invNorm(

invNorm( returns the x value that has a given area to its left.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Make sure the home screen is clear, then press 2ndDISTR to open the DISTR menu and choose 3:invNorm(.
2ndDISTR3
Fill in the wizard: area: .95, μ: 165.1 and σ: 6.35, pressing ▼︎ after each.
.95▼︎165.1▼︎6.35▼︎
Press ENTER on Paste to paste invNorm(.95,165.1,6.35) to the home screen.
ENTER
Press ENTER to evaluate. Only 5 percent of the women are taller than about 175.5 cm.
ENTER

Tutorial

Graph and shade the top 5 percent

ShadeNorm( draws the normal curve and shades the area between two bounds.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set the window: press WINDOW and enter Xmin = 145, Xmax = 185, Xscl = 5, Ymin = ⁻.02, Ymax = .08, Yscl = 0 and Xres = 1, pressing ENTER after each.
WINDOW145ENTER185ENTER5ENTER(−).02ENTER.08ENTER0ENTER1ENTER
Go back to the home screen.
2ndQUIT
Press 2ndDISTR ▶︎ to display DISTR DRAW and choose 1:ShadeNorm(.
2ndDISTR▶︎1
Enter the bounds and the curve: lower: 175.5448205 (the percentile you just found), upper: 1E99 (type 1 2ndEE 99, which stands for infinity), μ: 165.1 and σ: 6.35.
175.5448205▼︎12ndEE99▼︎165.1▼︎6.35▼︎
The cursor is on DRAW. Press ENTER to plot the curve and shade the upper 5 percent. The graph labels the area, the lower bound and the upper bound.
ENTER

Note The tour assumes STAT WIZARDS is on in MODE (the default). With it off, DISTR items paste the bare function name and you type the arguments yourself.

Inferential Stat Editorsp. 218

Choosing a hypothesis test or confidence interval from the STAT TESTS menu opens an inferential stat editor: a small form that asks for exactly what that test needs. Each editor is different, but they all work the same way.

Note ANOVA( is the one exception. It has no editor: choosing it pastes ANOVA( onto the home screen and you type the list names yourself.

Using an Inferential Stat Editorp. 218

Pick a hypothesis test or confidence interval from the STAT TESTS menu. Its editor opens.
If the editor offers Data or Stats input, choose one.
Enter a real number, list name or expression for every argument.
If offered, choose the alternative hypothesis (≠, < or >).
If offered, choose Yes or No for Pooled:.
Move to Calculate (or Draw, when it exists) and press ENTER.

Calculate shows the results on a scrolling results screen. Press ENTER or CLEAR to return home.
Draw plots the test statistic and p-value on a graph. The window is adjusted automatically to fit the curve.

The editors remember what you typed, so you can run several tests or intervals in a row without re-entering everything. The choices you highlight (Data or Stats, the alternative, Pooled) are remembered too.

Selecting Data or Statsp. 219

Most editors begin with Inpt: Data Stats.

Data means the sample lives in one or two lists. You are asked for the list names (and an optional frequency list).
Stats means you already know the summary statistics, such as x̄, Sx and n, and type them in directly.

To choose, move to the option with ◀︎ or ▶︎ and press ENTER. Some editors have no such choice: 1-PropZTest, 2-PropZTest, 1-PropZInt, 2-PropZInt, χ²-Test, χ²GOF-Test, LinRegTTest and LinRegTInt always take their inputs one way.

Entering the Values for Argumentsp. 219

Every row needs a value. A numeric row accepts a number or an expression; a list row accepts a list name, which you can paste from 2ndLIST (the list NAMES menu) or type with the alpha keys. If you are unsure what a symbol means, see Inferential Statistics Input Descriptions.

Type the value and press ENTER: it is stored and the cursor drops to the next row. ▼︎ and ▲︎ also move between rows.

Selecting an Alternative Hypothesis (≠ < >)p. 219

Most hypothesis-test editors have a row with three choices, one per alternative hypothesis:

≠ : two-sided, for example μ≠μ0 in Z-Test.
< : left-sided, for example μ1<μ2 in 2-SampTTest.
> : right-sided, for example p1>p2 in 2-PropZTest.

Move the highlight onto the one you want with ◀︎ or ▶︎, then press ENTER.

Tutorial

Run a right-sided test (μ>μ0)

Test whether the mean of L₁ exceeds 300, with σ = 3 known.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Z-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 1:Z-Test by pressing 1.
STAT▶︎▶︎1
Enter μ0 = 300 and σ = 3, pressing ENTER after each.
▼︎300ENTER3ENTER
Set List: to L₁ (2ndL1) and Freq: to 1.
2ndL1ENTER1ENTER
The cursor is on the alternative row (μ:≠μ0 <μ0 >μ0). Press ▶︎ twice to highlight >μ0, then press ENTER to select it.
▶︎▶︎ENTER
Move down to Calculate and press ENTER. The first result line now reads μ>300.
▼︎ENTER
Tidy up. Press ENTER to leave the results, reopen the editor, move down to the alternative row and press ◀︎ ◀︎ ENTER to put it back on ≠.
ENTERSTAT▶︎▶︎1▼︎▼︎▼︎▼︎▼︎◀︎◀︎ENTER

Selecting the Pooled Optionp. 220

Pooled: appears only in 2-SampTTest and 2-SampTInt. Choose No if the two population variances may differ (the default). Choose Yes if you are willing to assume the variances are equal; the two samples are then combined into a single pooled estimate, Sxp.

Highlight Yes or No and press ENTER.

Selecting Calculate or Draw for a Hypothesis Testp. 220

When all the arguments are filled in, a hypothesis test needs one last decision. Both options run immediately when you press ENTER:

Calculate lists the results (test statistic, p-value and so on) on a results screen.
Draw sketches the test statistic on the distribution curve, shades the p-value area, and prints the statistic and p-value under the graph. The window variables are set automatically.

Selecting Calculate for a Confidence Intervalp. 220

Confidence-interval editors have only Calculate; there is no Draw option. Press ENTER on it to see the interval and the statistics it was based on.

Bypassing the Inferential Stat Editorsp. 220

You can skip the editor by pasting the instruction itself from the CATALOG (2ndCATALOG) and typing its arguments on the home screen. This also works inside a program: choose the instruction from the CATALOG or from the STAT TESTS menu while editing the program.

The argument order for every instruction is in Appendix A. In general you give the inputs the editor would ask for, in the same order. The alternative is entered as ⁻1 for <, 0 for ≠ and 1 for >, and a final 1 asks for Draw instead of Calculate. For example, Z-Test(300,3,L₁,1,0) does the same as the Z-Test editor with Data input, List L₁, Freq 1 and ≠.

Note Whenever you compute a new test or interval, the output variables from the previous one are invalidated. See Test and Interval Output Variables.

STAT TESTS Menup. 221

Press STAT ▶︎ ▶︎ to open the TESTS menu. Picking an instruction opens its inferential stat editor (all but ANOVA(). Most instructions also store some results in output variables; see Test and Interval Output Variables.

1:Z-Test…	Test for one mean, σ known
2:T-Test…	Test for one mean, σ unknown
3:2-SampZTest…	Test comparing two means, σ’s known
4:2-SampTTest…	Test comparing two means, σ’s unknown
5:1-PropZTest…	Test for one proportion
6:2-PropZTest…	Test comparing two proportions
7:ZInterval…	Confidence interval for one mean, σ known
8:TInterval…	Confidence interval for one mean, σ unknown
9:2-SampZInt…	Confidence interval for the difference of two means, σ’s known
0:2-SampTInt…	Confidence interval for the difference of two means, σ’s unknown
A:1-PropZInt…	Confidence interval for one proportion
B:2-PropZInt…	Confidence interval for the difference of two proportions
C:χ²-Test…	Chi-square test for a two-way table
D:χ²GOF-Test…	Chi-square goodness-of-fit test
E:2-SampFTest…	Test comparing two standard deviations
F:LinRegTTest…	t test for the regression slope and r
G:LinRegTInt…	Confidence interval for the regression slope b
H:ANOVA(	One-way analysis of variance

Each instruction below lists the rows of its editor, the results it produces and a worked example. Where an instruction offers Data or Stats input, both are covered; the rows that belong only to Data or only to Stats stay hidden until you pick that input type. Where it offers Draw as well as Calculate, both outputs are covered.

Z-Testp. 223

Z-Test is a one-sample z test (STAT TESTS item 1). It tests whether the mean μ of a population equals a stated value μ0, when the population standard deviation σ is known. The null hypothesis is H0: μ=μ0; the alternative Ha is one of μ≠μ0, μ<μ0 or μ>μ0.

Z-Test

Z-Test(μ0,σ,list[,freq,alternative,drawflag])Z-Test(μ0,σ,x̄,n[,alternative,drawflag])

One-sample z test for a mean with known σ. Editor rows: Inpt (Data or Stats), μ0, σ, then either List and Freq or x̄ and n, the alternative, and Calculate / Draw. Calculate shows the alternative, z, p, x̄, Sx (Data only) and n.

Keys: STAT▶︎▶︎1

Z-Test(300,3,L₁,1,0) tests H0: μ=300 using the data in L₁ and a two-sided alternative.

The guidebook example tests the data L₁ = {299.4, 297.7, 301, 298.9, 300.2, 297} against μ0 = 300 with σ = 3. The tutorials below do exactly that: once with the list, once with summary statistics, and once drawing the result.

Z-Test
Inpt:Data Stats
μ0:300
σ:3
List:L₁
Freq:1
μ:≠μ0 <μ0 >μ0
Calculate

The Z-Test editor with Data input

Z-Test
Inpt:Data Stats
μ0:300
σ:3
x̄:299.0333
n:6
μ:≠μ0 <μ0 >μ0
Calculate

The same editor with Stats input

Z-Test
μ≠300
z=⁻.7892800282
p=.4299483568
x̄=299.0333333
Sx=1.502886112
n=6

Calculated results (Data input)

Note The guidebook prints its STAT TESTS examples in Fix 4 mode. In Float mode, as here, you see up to ten digits instead.

Tutorial

Z-Test with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Z-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 1:Z-Test by pressing 1.
STAT▶︎▶︎1
With Data selected, enter μ0 = 300 and σ = 3.
▼︎300ENTER3ENTER
Enter the list name L₁ (2ndL1) and a frequency of 1.
2ndL1ENTER1ENTER
Select the two-sided alternative ≠μ0 (already highlighted) by pressing ENTER.
ENTER
Move down to Calculate and press ENTER. The z statistic is about ⁻0.789 and the p-value about 0.430, so there is no evidence against H0: μ=300.
▼︎ENTER
Scroll down with ▼︎ to see the rest of the output: the sample mean x̄ = 299.0333, Sx = 1.5029 and n = 6.
▼︎▼︎

Tutorial

Z-Test with Stats input

The same test, supplying the sample mean 299.0333 and sample size 6 directly.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Z-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 1:Z-Test by pressing 1.
STAT▶︎▶︎1
Highlight Stats and select it. The List and Freq rows give way to x̄ and n.
▶︎ENTER
Enter μ0 = 300, σ = 3, x̄ = 299.0333 and n = 6.
▼︎300ENTER3ENTER299.0333ENTER6ENTER
Select the ≠μ0 alternative, move down to Calculate and press ENTER.
ENTER▼︎ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎1◀︎ENTER

Tutorial

Z-Test with Draw output

Draw plots the standard normal curve and shades the area beyond the test statistic.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the Z-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 1:Z-Test by pressing 1.
STAT▶︎▶︎1
Enter μ0 = 300, σ = 3, List L₁ and Freq 1, exactly as before.
▼︎300ENTER3ENTER2ndL1ENTER1ENTER
Select ≠μ0, then move down twice to Draw and press ENTER.
ENTER▼︎▼︎ENTER
The curve shows the shaded p-value region, and the z statistic and p-value are printed under the graph. Press 2ndQUIT (QUIT) when you are done to go to the home screen.
2ndQUIT

T-Testp. 224

T-Test is a one-sample t test (item 2). It tests a mean μ against μ0 when the population standard deviation is unknown, using the sample standard deviation Sx instead. H0 is μ=μ0 against μ≠μ0, μ<μ0 or μ>μ0.

T-Test

T-Test μ0[,listname,freq,alternative,drawflag]T-Test μ0,x̄,Sx,n[,alternative,drawflag]

One-sample t test for a mean with unknown σ. Editor rows: Inpt, μ0, then List and Freq or x̄, Sx and n, the alternative and Calculate / Draw. Results: the alternative, t, p, df, x̄, Sx and n.

Keys: STAT▶︎▶︎2

T-Test 100,98,12,6 tests μ=100 given x̄=98, Sx=12 and n=6.

The guidebook example uses the list TEST = {91.9, 97.8, 111.4, 122.3, 105.4, 95} with μ0 = 100.

Tutorial

T-Test with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the T-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 2:T-Test by pressing 2.
STAT▶︎▶︎2
Enter μ0 = 100.
▼︎100ENTER
Press 2ndLIST to open the list names menu and choose TEST (item 7), then press ENTER. Enter 1 for Freq:.
2ndLIST7ENTER1ENTER
Select ≠μ0, move down to Calculate and press ENTER.
ENTER▼︎ENTER

Tutorial

T-Test with Stats input

Supply the mean, standard deviation and sample size directly.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the T-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 2:T-Test by pressing 2.
STAT▶︎▶︎2
Select Stats input.
▶︎ENTER
Enter μ0 = 100, x̄ = 98, Sx = 12 and n = 6.
▼︎100ENTER98ENTER12ENTER6ENTER
Select ≠μ0, move down to Calculate and press ENTER.
ENTER▼︎ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎2◀︎ENTER

2-SampZTestp. 225

2-SampZTest (item 3) compares the means μ1 and μ2 of two populations using independent samples, when both population standard deviations σ1 and σ2 are known. H0 is μ1=μ2 against μ1≠μ2, μ1<μ2 or μ1>μ2.

2-SampZTest

2-SampZTest(σ1,σ2[,list1,list2,freq1,freq2,alternative,drawflag])2-SampZTest(σ1,σ2,x̄1,n1,x̄2,n2[,alternative,drawflag])

Two-sample z test for the difference of two means. Editor rows: Inpt, σ1, σ2, then List1, List2, Freq1, Freq2 (Data) or x̄1, n1, x̄2, n2 (Stats), the alternative, and Calculate / Draw. Results: the alternative, z, p, x̄1, x̄2, Sx1, Sx2 (Data only), n1 and n2.

Keys: STAT▶︎▶︎3

The guidebook example uses LISTA = {154, 109, 137, 115, 140} and LISTB = {108, 115, 126, 92, 146}, with σ1 = 15.5 and σ2 = 19.

Tutorial

2-SampZTest with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampZTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 3:2-SampZTest by pressing 3.
STAT▶︎▶︎3
Enter σ1 = 15.5 and σ2 = 19.
▼︎15.5ENTER19ENTER
Choose the two list names from the 2ndLIST menu (LISTA is item 7, LISTB item 8) and give both a frequency of 1.
2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER
Select ≠μ2, move down to Calculate and press ENTER. z is about 1.24 and p about 0.215, which is not significant.
ENTER▼︎ENTER

Tutorial

2-SampZTest with Stats input

Use the two sample means and sizes instead of the lists.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampZTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 3:2-SampZTest by pressing 3.
STAT▶︎▶︎3
Select Stats input.
▶︎ENTER
Enter σ1 = 15.5, σ2 = 19, x̄1 = 131, n1 = 5, x̄2 = 117.4 and n2 = 5.
▼︎15.5ENTER19ENTER131ENTER5ENTER117.4ENTER5ENTER
Select ≠μ2 and calculate. The result matches the Data version.
ENTER▼︎ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎3◀︎ENTER

2-SampTTestp. 226

2-SampTTest (item 4) compares two population means from independent samples when neither population standard deviation is known. H0 is μ1=μ2 against μ1≠μ2, μ1<μ2 or μ1>μ2. Use Pooled: Yes only if you can assume the two populations have equal variances.

2-SampTTest

2-SampTTest [list1,list2,freq1,freq2,alternative,pooled,drawflag]2-SampTTest x̄1,Sx1,n1,x̄2,Sx2,n2[,alternative,pooled,drawflag]

Two-sample t test for the difference of two means. Editor rows: Inpt, then List1, List2, Freq1, Freq2 (Data) or x̄1, Sx1, n1, x̄2, Sx2, n2 (Stats), the alternative, Pooled and Calculate / Draw. Results: the alternative, t, p, df, x̄1, x̄2, Sx1, Sx2, Sxp (only when pooled), n1 and n2.

Keys: STAT▶︎▶︎4

In the pooled case the degrees of freedom are n1+n2−2; otherwise they are computed from the two sample variances and are usually not a whole number.

The guidebook example uses SAMP1 = {12.207, 16.869, 25.05, 22.429, 8.456, 10.589} and SAMP2 = {11.074, 9.686, 12.064, 9.351, 8.182, 6.642}.

Tutorial

2-SampTTest with Data input (not pooled)

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampTTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 4:2-SampTTest by pressing 4.
STAT▶︎▶︎4
With Data selected, move down and choose SAMP1 (item 7 in 2ndLIST) for List1: and SAMP2 (item 8) for List2:. Give both frequencies the value 1.
▼︎2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER
Select ≠μ2, then move down to Pooled: and select No.
ENTER▼︎ENTER
Move down to Calculate and press ENTER. t is about 2.258 with df ≈ 5.84 and p ≈ 0.066.
▼︎ENTER

Tutorial

2-SampTTest with pooled variances

Assuming equal variances gives the same t, but more degrees of freedom and a smaller p-value.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampTTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 4:2-SampTTest by pressing 4.
STAT▶︎▶︎4
Choose SAMP1 and SAMP2 for the lists and use a frequency of 1 for each.
▼︎2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER
Select ≠μ2, move down to Pooled:, highlight Yes with ▶︎ and press ENTER.
ENTER▼︎▶︎ENTER
Move down to Calculate and press ENTER. df is now n1+n2−2 = 10 and the pooled standard deviation Sxp appears.
▼︎ENTER
Tidy up. Press ENTER to leave the results, reopen the editor, move down to Pooled: and press ◀︎ ENTER to put it back on No.
ENTERSTAT▶︎▶︎4▼︎▼︎▼︎▼︎▼︎▼︎◀︎ENTER

Tutorial

2-SampTTest with Stats input

The same two samples described by their summary statistics.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampTTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 4:2-SampTTest by pressing 4.
STAT▶︎▶︎4
Select Stats input.
▶︎ENTER
Enter x̄1 = 15.9333, Sx1 = 6.7014, n1 = 6, x̄2 = 9.4998, Sx2 = 1.9501 and n2 = 6.
▼︎15.9333ENTER6.7014ENTER6ENTER9.4998ENTER1.9501ENTER6ENTER
Select ≠μ2 and Pooled: No, then calculate.
ENTER▼︎ENTER▼︎ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎4◀︎ENTER

1-PropZTestp. 227

1-PropZTest (item 5) tests an unknown proportion of successes. You give the number of successes x and the number of observations n. H0 is prop=p0 against prop≠p0, prop<p0 or prop>p0.

1-PropZTest

1-PropZTest(p0,x,n[,alternative,drawflag])

One-proportion z test. Editor rows: p0, x, n, the alternative, and Calculate / Draw. Results: the alternative, z, p, p̂ (the sample proportion x/n) and n. There is no Data/Stats choice.

Keys: STAT▶︎▶︎5

Tutorial

1-PropZTest: is a coin fair?

In 4040 tosses a coin shows 2048 heads. Test H0: prop = 0.5.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 1-PropZTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 5:1-PropZTest by pressing 5.
STAT▶︎▶︎5
Enter p0 = .5, x = 2048 and n = 4040.
.5ENTER2048ENTER4040ENTER
Select ≠p0, move down to Calculate and press ENTER. The sample proportion is about 0.5069 and p is about 0.378, so the coin looks fair.
ENTER▼︎ENTER

2-PropZTestp. 228

2-PropZTest (item 6) compares the proportions of successes p1 and p2 in two populations. You give the successes x1 and x2 and the sample sizes n1 and n2. H0 is p1=p2, tested with the pooled sample proportion p̂, against p1≠p2, p1<p2 or p1>p2.

2-PropZTest

2-PropZTest(x1,n1,x2,n2[,alternative,drawflag])

Two-proportion z test. Editor rows: x1, n1, x2, n2, the alternative, and Calculate / Draw. Results: the alternative, z, p, p̂1, p̂2, the pooled p̂, n1 and n2.

Keys: STAT▶︎▶︎6

Tutorial

2-PropZTest: two groups of patients

Group 1 has 49 successes out of 61; group 2 has 56 out of 62.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-PropZTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 6:2-PropZTest by pressing 6.
STAT▶︎▶︎6
Enter x1 = 49, n1 = 61, x2 = 56 and n2 = 62.
49ENTER61ENTER56ENTER62ENTER
Select ≠p2, move down to Calculate and press ENTER.
ENTER▼︎ENTER

Tutorial

2-PropZTest with Draw output

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-PropZTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 6:2-PropZTest by pressing 6.
STAT▶︎▶︎6
Enter the same counts: x1 = 49, n1 = 61, x2 = 56, n2 = 62.
49ENTER61ENTER56ENTER62ENTER
Select ≠p2, then move down twice to Draw and press ENTER. The normal curve is drawn with the p-value area shaded.
ENTER▼︎▼︎ENTER
Press 2ndQUIT (QUIT) to return to the home screen.
2ndQUIT

ZIntervalp. 229

ZInterval (item 7) computes a confidence interval for a population mean μ when the population standard deviation σ is known. The width of the interval depends on the C-Level you choose.

ZInterval

ZInterval σ,list[,freq,CLevel]ZInterval σ,x̄,n[,CLevel]

Confidence interval for a mean with known σ. Editor rows: Inpt, σ, then List and Freq or x̄ and n, C-Level and Calculate. Results: the interval (lower, upper), x̄, Sx (Data only) and n.

Keys: STAT▶︎▶︎7

The guidebook example uses L₁ = {299.4, 297.7, 301, 298.9, 300.2, 297} with σ = 3 and a 99 percent confidence level.

Tutorial

ZInterval with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the ZInterval editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 7:ZInterval by pressing 7.
STAT▶︎▶︎7
With Data selected enter σ = 3.
▼︎3ENTER
Enter List L₁ (2ndL1), Freq 1 and a C-Level of .99.
2ndL1ENTER1ENTER.99ENTER
The cursor has moved to Calculate. Press ENTER. The 99 percent interval is about (295.88, 302.19).
ENTER

Tutorial

ZInterval with Stats input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the ZInterval editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 7:ZInterval by pressing 7.
STAT▶︎▶︎7
Select Stats input.
▶︎ENTER
Enter σ = 3, x̄ = 299.0333, n = 6 and a C-Level of .99.
▼︎3ENTER299.0333ENTER6ENTER.99ENTER
Press ENTER on Calculate.
ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎7◀︎ENTER

TIntervalp. 229

TInterval (item 8) computes a confidence interval for a population mean μ when σ is unknown, using the sample standard deviation and the t distribution. Getting Started walks through an example with heights.

TInterval

TInterval [list,freq,CLevel]TInterval x̄,Sx,n[,CLevel]

Confidence interval for a mean with unknown σ. Editor rows: Inpt, then List and Freq or x̄, Sx and n, C-Level and Calculate. Results: the interval (lower, upper), x̄, Sx and n.

Keys: STAT▶︎▶︎8

The guidebook example uses L₆ = {1.6, 1.7, 1.8, 1.9}.

Tutorial

TInterval with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the TInterval editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 8:TInterval by pressing 8.
STAT▶︎▶︎8
Enter List L₆ (2ndL6), a frequency of 1 and a C-Level of .95.
▼︎2ndL6ENTER1ENTER.95ENTER
Press ENTER on Calculate. The 95 percent interval for the mean is about (1.545, 1.955), from x̄ = 1.75, Sx ≈ 0.129 and n = 4.
ENTER

2-SampZIntp. 230

2-SampZInt (item 9) gives a confidence interval for the difference μ1−μ2 of two population means when both σ1 and σ2 are known.

2-SampZInt

2-SampZInt(σ1,σ2[,list1,list2,freq1,freq2,CLevel])2-SampZInt(σ1,σ2,x̄1,n1,x̄2,n2[,CLevel])

Confidence interval for the difference of two means with known σ1 and σ2. Editor rows: Inpt, σ1, σ2, then List1, List2, Freq1, Freq2 or x̄1, n1, x̄2, n2, C-Level and Calculate. Results: the interval, x̄1, x̄2, Sx1, Sx2 (Data only), n1 and n2.

Keys: STAT▶︎▶︎9

The guidebook example uses LISTC = {154, 109, 137, 115, 140} and LISTD = {108, 115, 126, 92, 146}, with σ1 = 15.5 and σ2 = 19.

Tutorial

2-SampZInt with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampZInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 9:2-SampZInt by pressing 9.
STAT▶︎▶︎9
Enter σ1 = 15.5 and σ2 = 19.
▼︎15.5ENTER19ENTER
Choose LISTC (item 7 in 2ndLIST) and LISTD (item 8), give each a frequency of 1, and enter a C-Level of .9.
2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER.9ENTER
Press ENTER on Calculate. The 90 percent interval for μ1−μ2 is about (⁻4.44, 31.64); it contains 0.
ENTER

Tutorial

2-SampZInt with Stats input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampZInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 9:2-SampZInt by pressing 9.
STAT▶︎▶︎9
Select Stats input.
▶︎ENTER
Enter σ1 = 15.5, σ2 = 19, x̄1 = 131, n1 = 5, x̄2 = 117.4, n2 = 5 and a C-Level of .9.
▼︎15.5ENTER19ENTER131ENTER5ENTER117.4ENTER5ENTER.9ENTER
Press ENTER on Calculate.
ENTER
Tidy up. The editor remembers its choices, so press ENTER to leave the results, reopen the editor and press ◀︎ ENTER to put Inpt: back on Data.
ENTERSTAT▶︎▶︎9◀︎ENTER

2-SampTIntp. 231

2-SampTInt (item 0) gives a confidence interval for the difference μ1−μ2 of two means when both population standard deviations are unknown. As in 2-SampTTest, choose Pooled: Yes only if you can assume equal variances.

2-SampTInt

2-SampTInt [list1,list2,freq1,freq2,CLevel,pooled]2-SampTInt x̄1,Sx1,n1,x̄2,Sx2,n2[,CLevel,pooled]

Confidence interval for the difference of two means with unknown σ. Editor rows: Inpt, then List1, List2, Freq1, Freq2 or x̄1, Sx1, n1, x̄2, Sx2, n2, C-Level, Pooled and Calculate. Results: the interval, df, x̄1, x̄2, Sx1, Sx2, Sxp (only when pooled), n1 and n2.

Keys: STAT▶︎▶︎0

The guidebook example uses SAMP1 = {12.207, 16.869, 25.05, 22.429, 8.456, 10.589} and SAMP2 = {11.074, 9.686, 12.064, 9.351, 8.182, 6.642}.

Tutorial

2-SampTInt with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampTInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose 0:2-SampTInt by pressing 0.
STAT▶︎▶︎0
Choose SAMP1 and SAMP2 from the 2ndLIST menu, give both frequencies the value 1, and enter a C-Level of .95.
▼︎2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER.95ENTER
Select Pooled: No, then move down to Calculate.
ENTER▼︎ENTER

1-PropZInt and 2-PropZIntp. 232

1-PropZInt (item A) gives a confidence interval for an unknown proportion of successes, from the success count x and the sample size n. 2-PropZInt (item B) gives a confidence interval for the difference p1−p2 between the proportions in two populations, from x1, n1, x2 and n2. Neither has a Data/Stats choice.

1-PropZInt

1-PropZInt x,n[,CLevel]

One-proportion z confidence interval. Editor rows: x, n, C-Level and Calculate. Results: the interval, p̂ and n.

Keys: STAT▶︎▶︎ALPHAA

2-PropZInt

2-PropZInt x1,n1,x2,n2[,CLevel]

Two-proportion z confidence interval for p1−p2. Editor rows: x1, n1, x2, n2, C-Level and Calculate. Results: the interval, p̂1, p̂2, n1 and n2.

Keys: STAT▶︎▶︎ALPHAB

Tutorial

1-PropZInt: 8 successes in 50 trials

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 1-PropZInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose A:1-PropZInt by pressing ALPHAA.
STAT▶︎▶︎ALPHAA
Enter x = 8, n = 50 and a C-Level of .95.
8ENTER50ENTER.95ENTER
Press ENTER on Calculate. p̂ = 0.16 and the 95 percent interval is about (0.058, 0.262).
ENTER

Tutorial

2-PropZInt: two groups of patients

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-PropZInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose B:2-PropZInt by pressing ALPHAB.
STAT▶︎▶︎ALPHAB
Enter x1 = 49, n1 = 61, x2 = 56, n2 = 62 and a C-Level of .99.
49ENTER61ENTER56ENTER62ENTER.99ENTER
Press ENTER on Calculate. The 99 percent interval for p1−p2 is about (⁻0.263, 0.063).
ENTER

χ²-Testp. 233

χ²-Test (item C) is a chi-square test for association in a two-way table of counts. H0 says there is no association between the row variable and the column variable; the alternative says the variables are related.

First store the observed counts in a matrix (at least 2×2, all whole numbers ≥ 0). The editor asks for Observed: (default [A]) and Expected: (default [B]). Expected is the matrix where the calculator saves the expected counts it works out. Results: χ², p and df.

χ²-Test

χ²-Test(observedmatrix,expectedmatrix[,drawflag])

Chi-square test for association on a two-way table. Editor rows: Observed, Expected, and Calculate / Draw. The expected counts are written to the Expected matrix. Results: χ², p and df.

Keys: STAT▶︎▶︎ALPHAC

χ²-Test([A],[B]) tests the table in [A] and stores the expected counts in [B].

The tutorials use this 2×3 table of observed counts, a made-up example: row 1 is 23, 15, 32 and row 2 is 47, 41, 16.

Tutorial

Enter the observed counts in matrix [A]

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the matrix editor for [A]: press 2ndMATRIX (MATRIX), move to EDIT with ▶︎ ▶︎ and choose 1:[A].
2ndMATRIX▶︎▶︎1
Set the size to 2 rows by 3 columns, then type the six counts row by row, pressing ENTER after each: 23, 15, 32, 47, 41, 16.
2ENTER3ENTER23ENTER15ENTER32ENTER47ENTER41ENTER16ENTER
Quit to the home screen.
2ndQUIT

Tutorial

Run χ²-Test and view the expected matrix

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the χ²-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose C:χ²-Test by pressing ALPHAC.
STAT▶︎▶︎ALPHAC
Make sure Observed: is [A] and Expected: is [B]. If you need to change them, press 2ndMATRIX to open the matrix names menu and pick the matrix. Here, pick [A] for Observed and [B] for Expected.
2ndMATRIX1ENTER2ndMATRIX2ENTER
The cursor is on Calculate. Press ENTER. χ² is about 19.74 with df = 2, and p is very small (about 5.2×10⁻⁵): strong evidence of association.
ENTER
Press ENTER to return home. The expected counts were stored in [B].
ENTER
To look at the matrix, press 2ndMATRIX, move to [B] with ▼︎ and press ENTER to paste it, then ENTER to display it.
2ndMATRIX▼︎ENTERENTER

Tutorial

χ²-Test with Draw output

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the χ²-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose C:χ²-Test by pressing ALPHAC.
STAT▶︎▶︎ALPHAC
Pick [A] and [B] again for the two matrices.
2ndMATRIX1ENTER2ndMATRIX2ENTER
Move down to Draw and press ENTER. The χ² density curve is drawn with the area beyond the statistic shaded.
▼︎ENTER

χ²GOF-Testp. 234

χ²GOF-Test (item D) is the chi-square goodness-of-fit test. It checks whether sample counts agree with the counts expected from a specified distribution. The editor asks for Observed: (a list of counts), Expected: (a list of the expected counts, same length) and df:, the degrees of freedom.

The degrees of freedom are the number of categories, minus one more than the number of parameters you estimated from the data. For example, with five categories and no estimated parameters, df = 5 − 1 = 4. Besides χ², p and df, the test stores a list named CNTRB holding each category's contribution to χ².

χ²GOF-Test

χ²GOF-Test(observedlist,expectedlist,df[,drawflag])

Chi-square goodness-of-fit test. Editor rows: Observed, Expected, df, and Calculate / Draw. Results: χ², p, df and the list CNTRB of per-category contributions.

Keys: STAT▶︎▶︎ALPHAD

The guidebook example uses Observed = {16, 25, 22, 8, 10} and Expected = {16.2, 21.6, 16.2, 14.4, 12.6}, with df = 4.

Tutorial

χ²GOF-Test

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the χ²GOF-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose D:χ²GOF-Test by pressing ALPHAD.
STAT▶︎▶︎ALPHAD
Enter L₁ for Observed: and L₂ for Expected:.
2ndL1ENTER2ndL2ENTER
Enter df = 4.
4ENTER
The cursor is on Calculate. Press ENTER. χ² is about 5.995 and p about 0.200, so these counts are consistent with the expected ones.
ENTER

Tutorial

χ²GOF-Test with Draw output

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the χ²GOF-Test editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose D:χ²GOF-Test by pressing ALPHAD.
STAT▶︎▶︎ALPHAD
Enter L₁, L₂ and df = 4.
2ndL1ENTER2ndL2ENTER4ENTER
Move down to Draw and press ENTER.
▼︎ENTER

2-SampFTestp. 235

2-SampFTest (item E) compares the standard deviations σ1 and σ2 of two normal populations whose means and standard deviations are all unknown. It uses the ratio of sample variances Sx1²/Sx2². H0 is σ1=σ2 against σ1≠σ2, σ1<σ2 or σ1>σ2.

2-SampFTest

2-SampFTest [list1,list2,freq1,freq2,alternative,drawflag]2-SampFTest Sx1,n1,Sx2,n2[,alternative,drawflag]

Two-sample F test for equal standard deviations. Editor rows: Inpt, then List1, List2, Freq1, Freq2 (Data) or Sx1, n1, Sx2, n2 (Stats), the alternative and Calculate / Draw. Results: the alternative, F, p, Sx1, Sx2, x̄1 and x̄2 (Data only), n1 and n2.

Keys: STAT▶︎▶︎ALPHAE

The guidebook example uses SAMP4 = {7, ⁻4, 18, 17, ⁻3, ⁻5, 1, 10, 11, ⁻2} and SAMP5 = {⁻1, 12, ⁻1, ⁻3, 3, ⁻5, 5, 2, ⁻11, ⁻1, ⁻3}.

Tutorial

2-SampFTest with Data input

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampFTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose E:2-SampFTest by pressing ALPHAE.
STAT▶︎▶︎ALPHAE
With Data selected, choose SAMP4 (item 7 in 2ndLIST) for List1: and SAMP5 (item 8) for List2:, with frequencies of 1.
▼︎2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER
Select σ1≠σ2, move down to Calculate and press ENTER. F ≈ 2.196 with p ≈ 0.236.
ENTER▼︎ENTER

Tutorial

2-SampFTest with Draw output

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the 2-SampFTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose E:2-SampFTest by pressing ALPHAE.
STAT▶︎▶︎ALPHAE
Choose SAMP4 and SAMP5 again, with frequencies of 1.
▼︎2ndLIST7ENTER2ndLIST8ENTER1ENTER1ENTER
Select σ1≠σ2, then move down twice to Draw and press ENTER. The F density curve is drawn with the p-value area shaded.
ENTER▼︎▼︎ENTER

LinRegTTestp. 236

LinRegTTest (item F) fits the line y=a+bx to your data and runs a t test on the slope b (equivalently, the correlation r). H0 is b=0 (and r=0) against b&r≠0, b&r<0 or b&r>0.

The regression equation is stored in RegEQ (VARS Statistics, EQ). If you type a Y= variable such as Y₁ at the RegEQ: prompt, the equation is also stored there and that function is turned on. The test creates a list of residuals named RESID, which then appears on the list NAMES menu.

Note In Fix mode the stored equation has only the fixed number of decimal places, which can reduce the accuracy of the fit if you choose very few digits.

LinRegTTest

LinRegTTest [Xlist,Ylist,freq,alternative,RegEQ]

Linear regression t test. Editor rows: Xlist, Ylist, Freq, the alternative (b&r), RegEQ and Calculate. Results: y=a+bx, the alternative, t, p, df, a, b, s (standard error about the line), r² and r.

Keys: STAT▶︎▶︎ALPHAF

The guidebook example uses L₃ = {38, 56, 59, 64, 74} and L₄ = {41, 63, 70, 72, 84}, storing the equation in Y₁.

Tutorial

LinRegTTest, storing the equation in Y₁

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the LinRegTTest editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose F:LinRegTTest by pressing ALPHAF.
STAT▶︎▶︎ALPHAF
Enter L₃ for Xlist:, L₄ for Ylist: and 1 for Freq:.
2ndL3ENTER2ndL4ENTER1ENTER
Select the two-sided alternative ≠0, then move down to the RegEQ: row.
ENTER▼︎
Paste Y₁ with VARS ▶︎ (the Y-VARS menu), then 1:Function… and 1:Y₁.
VARS▶︎11
Press ENTER to move onto Calculate, then ENTER again. The slope b ≈ 1.197 and r ≈ 0.994.
ENTERENTER
Scroll down to see s, r² and r. The residuals are in the list RESID.
▼︎▼︎▼︎

LinRegTIntp. 237

LinRegTInt (item G) computes a confidence interval for the slope b of the regression line. If the interval contains 0, the data do not give enough evidence of a linear relationship.

Xlist: and Ylist: are the lists of independent and dependent values (same length).
Freq: is a list of frequencies for each data point; the default is 1 (each point counted once). Entries must be real numbers.
C-Level: is the confidence level; the default is .95.
RegEQ: (optional) is a Y= variable in which to store the regression equation.

LinRegTInt

LinRegTInt [Xlist,Ylist,freq,CLevel,RegEQ]

Confidence interval for the slope of the regression line. Editor rows: Xlist, Ylist, Freq, C-Level, RegEQ and Calculate. Results: y=a+bx, the interval for b, b, df, s, a, r² and r.

Keys: STAT▶︎▶︎ALPHAG

The guidebook example uses list 1 = {4, 5, 6, 7, 8} and list 2 = {1, 2, 3, 3.5, 4.5}.

Tutorial

LinRegTInt

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Open the LinRegTInt editor: press STAT, move to TESTS with ▶︎ ▶︎, then choose G:LinRegTInt by pressing ALPHAG.
STAT▶︎▶︎ALPHAG
Enter L₁ for Xlist:, L₂ for Ylist:, a frequency of 1 and a C-Level of .95.
2ndL1ENTER2ndL2ENTER1ENTER.95ENTER
Leave RegEQ: blank: press CLEAR to empty the field and ENTER to move on.
CLEARENTER
Press ENTER on Calculate. The interval for the slope is about (0.691, 1.009), from b = 0.85 with df = 3.
ENTER

ANOVA(p. 238

ANOVA( (item H) is a one-way analysis of variance for comparing the means of two to 20 populations. It looks at how much the group means vary compared with the variation inside each group. H0 is μ1=μ2=…=μk against the alternative that not all the means are equal. SS means sum of squares and MS means mean square.

ANOVA(

ANOVA(list1,list2[,…,list20])

One-way analysis of variance on two to twenty lists. There is no editor: the instruction is pasted on the home screen. Results: F, p, then Factor (df, SS, MS) and Error (df, SS, MS) and the pooled standard deviation Sxp.

Keys: STAT▶︎▶︎ALPHAH

ANOVA(L₁,L₂,L₃)

The guidebook example uses L₁ = {7, 4, 6, 6, 5}, L₂ = {6, 5, 5, 8, 7} and L₃ = {4, 7, 6, 7, 6}.

Tutorial

One-way ANOVA of three groups

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Choose H:ANOVA( from the STAT TESTS menu. It is pasted to the home screen. (H is the alpha letter on MATH.)
STAT▶︎▶︎ALPHAH
Type the three list names separated by commas, then close the parenthesis.
2ndL1,2ndL2,2ndL3)
Press ENTER. F ≈ 0.311 and p ≈ 0.738: no evidence that the group means differ. Scroll with ▼︎ to see the Factor and Error rows.
ENTER
Scroll down to see the Error row and the pooled standard deviation.
▼︎▼︎▼︎▼︎

Inferential Statistics Input Descriptionsp. 239

These are the inputs the inferential stat editors ask for, in the order they appear in this chapter.

Input	Meaning
μ0	The hypothesized value of the population mean being tested.
σ	The known population standard deviation. Must be a real number > 0.
List	Name of the list holding the data being tested.
Freq	Name of a list of frequencies for the values in List. Default 1. Every entry must be an integer ≥ 0.
Calculate / Draw	The kind of output. Calculate shows results on a results screen; for tests, Draw graphs them.
x̄, Sx, n	Summary statistics (mean, standard deviation, size) for the one-sample tests and intervals.
σ1	Known standard deviation of the first population, for two-sample tests and intervals. Must be > 0.
σ2	Known standard deviation of the second population. Must be > 0.
List1, List2	Names of the two data lists for two-sample tests and intervals. Defaults L₁ and L₂.
Freq1, Freq2	Frequency lists for List1 and List2. Defaults 1. Entries must be integers ≥ 0.
x̄1, Sx1, n1, x̄2, Sx2, n2	Summary statistics for sample one and sample two in the two-sample tests and intervals.
Pooled	Used by 2-SampTTest and 2-SampTInt. No does not pool the variances; Yes does.
p0	The expected sample proportion for 1-PropZTest. Must satisfy 0 < p0 < 1.
x	The count of successes in the sample, for 1-PropZTest and 1-PropZInt. An integer ≥ 0.
n	The number of observations in the sample, for 1-PropZTest and 1-PropZInt. An integer > 0.
x1, x2	Counts of successes in samples one and two, for 2-PropZTest and 2-PropZInt. Integers ≥ 0.
n1, n2	Numbers of observations in samples one and two, for 2-PropZTest and 2-PropZInt. Integers > 0.
C-Level	The confidence level for interval instructions. Must be ≥ 0 and < 100. A value ≥ 1 is read as a percentage and divided by 100. Default .95.
Observed (matrix)	The matrix of observed counts of a two-way table, for χ²-Test (and the observed list in χ²GOF-Test). All entries integers ≥ 0; at least 2×2 for a matrix.
Expected (matrix)	Where the expected counts are stored. It is created when the test completes successfully.
df	Degrees of freedom for χ²GOF-Test: the number of categories minus one more than the number of estimated parameters.
Xlist, Ylist	Lists holding the data for LinRegTTest and LinRegTInt. Defaults L₁ and L₂; they must be the same length.
RegEQ	The Y= variable in which to store the regression equation, which is then turned on. Default: the equation is stored only in the RegEQ variable.

Test and Interval Output Variablesp. 240

After a test or interval runs, its results are saved in variables you can use in expressions. Press VARS, choose 5:Statistics, and open the submenu named in the last column. Computing a new test or interval invalidates all the previous output variables.

Variable	Tests	Intervals	LinRegTTest, ANOVA	VARS menu
p-value	p		p	TEST
test statistic	z, t, χ², F		t, F	TEST
degrees of freedom	df	df	df	TEST
sample mean of x, samples 1 and 2	x̄1, x̄2	x̄1, x̄2		TEST
sample standard deviation of x, samples 1 and 2	Sx1, Sx2	Sx1, Sx2		TEST
number of data points, samples 1 and 2	n1, n2	n1, n2		TEST
pooled standard deviation	Sxp	Sxp	Sxp	TEST
estimated sample proportion	p̂	p̂		TEST
estimated proportion, population 1	p̂1	p̂1		TEST
estimated proportion, population 2	p̂2	p̂2		TEST
confidence interval pair		lower, upper		TEST
mean of x values	x̄	x̄		XY
sample standard deviation of x	Sx	Sx		XY
number of data points	n	n		XY
standard error about the line			s	TEST
regression / fit coefficients			a, b	EQ
correlation coefficient			r	EQ
coefficient of determination			r²	EQ
regression equation			RegEQ	EQ

Note These variables cannot be archived.

Distribution Functionsp. 241

Press 2ndDISTR to open the DISTR menu. It has two tabs, DISTR (probability functions) and DRAW (shading, covered in Distribution Shading).

1:normalpdf(	Normal probability density function
2:normalcdf(	Normal cumulative distribution function
3:invNorm(	Inverse cumulative normal distribution
4:invT(	Inverse cumulative Student-t distribution
5:tpdf(	Student-t probability density
6:tcdf(	Student-t distribution probability
7:χ²pdf(	Chi-square probability density
8:χ²cdf(	Chi-square distribution probability
9:Fpdf(	F probability density
0:Fcdf(	F distribution probability
A:binompdf(	Binomial probability
B:binomcdf(	Binomial cumulative probability
C:poissonpdf(	Poisson probability
D:poissoncdf(	Poisson cumulative probability
E:geometpdf(	Geometric probability
F:geometcdf(	Geometric cumulative probability

Note With STAT WIZARDS on in MODE (the default), choosing any DISTR function opens a small input screen (a wizard) asking for each argument by name. Fill the fields, move to Paste and press ENTER to paste the finished call; press ENTER again to run it. Optional arguments already show their defaults. With STAT WIZARDS off, the bare function name is pasted instead and you type the arguments yourself.

Note In the Y= editor or a program, the wizard pastes the function there instead of running it.

normalpdf(p. 242

normalpdf(

normalpdf(x[,μ,σ])

Probability density function of the normal distribution at x. The defaults are μ=0 and σ=1. Paste it into the Y= editor to plot the bell curve. The density is f(x)=1/(σ√(2π))·e^(−(x−μ)²/(2σ²)), with σ>0.

Keys: 2ndDISTR1

normalpdf(0) → .3989422804
normalpdf(35,35,2) → .1994711402

Note To plot the curve, set Xmin and Xmax so that the mean μ falls between them, then choose 0:ZoomFit from the ZOOM menu.

Tutorial

Evaluate normalpdf(

Height of the standard normal curve at x = 1.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 1:normalpdf( by pressing 1. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR1
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
1▼︎▼︎▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate normalpdf(1). The result is .2419707245.
ENTER

Tutorial

Plot a normal curve in Y=

Graph a normal distribution with mean 35 and standard deviation 2.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set the window so the whole curve fits: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW28ENTER42ENTER1ENTER0ENTER.2ENTER.1ENTER
Open the Y= editor with the cursor on Y₁.
Y=
Press 2ndDISTR and choose 1:normalpdf( by pressing 1. Because you are in the Y= editor, the wizard will paste the function into the equation.
2ndDISTR1
The first field is the x value: press X,T,θ,n to enter X. Fill in the other fields and press ▼︎ after each.
X,T,θ,n▼︎35▼︎2▼︎
Press ENTER on Paste to place the function into Y₁.
ENTER
Press GRAPH to draw the curve. For the function normalpdf(X,35,2), Y₁(35) = 0.1994711402.
GRAPH

normalcdf(p. 242

normalcdf(

normalcdf(lowerbound,upperbound[,μ,σ])

Probability that a normal random variable lies between lowerbound and upperbound. The defaults are μ=0 and σ=1.

Keys: 2ndDISTR2

normalcdf(⁻1,1) → .6826894921
normalcdf(100,1E99,90,15) → the chance of exceeding 100 when μ=90, σ=15

Note To mean infinity, type 1E99 (12ndEE99) for an upper bound, or ⁻1E99 for a lower bound. For example, to see the area to the left of an upper bound, make the lower bound ⁻1E99.

Tutorial

Probability within one standard deviation

About 68 percent of a normal distribution lies within 1 standard deviation of the mean.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 2:normalcdf( by pressing 2. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR2
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
(−)1▼︎1▼︎▼︎▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate normalcdf(⁻1,1). The result is .6826894921.
ENTER

invNorm(p. 243

invNorm(

invNorm(area[,μ,σ])

Inverse of the cumulative normal distribution: returns the x value that has the given area to its left under the normal curve with mean μ and standard deviation σ. The area must satisfy 0 ≤ area ≤ 1. The defaults are μ=0 and σ=1.

Keys: 2ndDISTR3

invNorm(.975) → 1.95996398
invNorm(.95,165.1,6.35) → 175.5448205

Tutorial

The z value with 97.5 percent of the area to its left

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 3:invNorm( by pressing 3. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR3
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
.975▼︎▼︎▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate invNorm(.975). The result is 1.95996398.
ENTER

invT(p. 243

invT(

invT(area,df)

Inverse of the cumulative Student-t distribution: returns the x value that has the given area to its left under the t curve with df degrees of freedom.

Keys: 2ndDISTR4

invT(.975,10) → 2.228138852

Tutorial

A t critical value

The 97.5th percentile of the t distribution with 10 degrees of freedom, used for a 95 percent interval with n = 11.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 4:invT( by pressing 4. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR4
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
.975▼︎10▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate invT(.975,10). The result is 2.228138852.
ENTER

tpdf(p. 243

tpdf(

tpdf(x,df)

Probability density function of the Student-t distribution at x. df (degrees of freedom) must be > 0. Paste it into Y= to plot the curve. The density is f(x)=Γ((df+1)/2)/(√(π·df)·Γ(df/2))·(1+x²/df)^(−(df+1)/2).

Keys: 2ndDISTR5

tpdf(1,10) → .2303619892

Note A good window for plotting: Xmin=⁻4.5, Xmax=4.5, Ymin=0, Ymax=.4.

Tutorial

Evaluate tpdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 5:tpdf( by pressing 5. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR5
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
1▼︎10▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate tpdf(1,10). The result is .2303619892.
ENTER

Tutorial

Plot a Student-t curve in Y=

Graph the t density with 2 degrees of freedom.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set the window so the whole curve fits: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW(−)4.5ENTER4.5ENTER1ENTER0ENTER.4ENTER.1ENTER
Open the Y= editor with the cursor on Y₁.
Y=
Press 2ndDISTR and choose 5:tpdf( by pressing 5. Because you are in the Y= editor, the wizard will paste the function into the equation.
2ndDISTR5
The first field is the x value: press X,T,θ,n to enter X. Fill in the other fields and press ▼︎ after each.
X,T,θ,n▼︎2▼︎
Press ENTER on Paste to place the function into Y₁.
ENTER
Press GRAPH to draw the curve. For the function tpdf(X,2), Y₁(0) = 0.3535533906.
GRAPH

tcdf(p. 244

tcdf(

tcdf(lowerbound,upperbound,df)

Probability that a Student-t variable with df degrees of freedom (df > 0) lies between lowerbound and upperbound.

Keys: 2ndDISTR6

tcdf(⁻2,2,10) → .9266119652

Note To mean infinity, type 1E99 (12ndEE99) for an upper bound, or ⁻1E99 for a lower bound. For example, to see the area to the left of an upper bound, make the lower bound ⁻1E99.

Tutorial

Evaluate tcdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 6:tcdf( by pressing 6. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR6
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
(−)2▼︎2▼︎10▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate tcdf(⁻2,2,10). The result is .9266119652.
ENTER

χ²pdf(p. 244

χ²pdf(

χ²pdf(x,df)

Probability density function of the χ² (chi-square) distribution at x ≥ 0. df must be an integer > 0. Paste it into Y= to plot the curve. The density is f(x)=(1/2)^(df/2)/Γ(df/2) · x^(df/2−1)·e^(−x/2).

Keys: 2ndDISTR7

χ²pdf(3,4) → .1673476201

Note A good window for plotting: Xmin=0, Xmax=30, Ymin=⁻.02, Ymax=.132.

Tutorial

Evaluate χ²pdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 7:χ²pdf( by pressing 7. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR7
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
3▼︎4▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate χ²pdf(3,4). The result is .1673476201.
ENTER

Tutorial

Plot a χ² curve in Y=

Graph the chi-square density with 9 degrees of freedom.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set the window so the whole curve fits: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW0ENTER30ENTER5ENTER(−).02ENTER.132ENTER.05ENTER
Open the Y= editor with the cursor on Y₁.
Y=
Press 2ndDISTR and choose 7:χ²pdf( by pressing 7. Because you are in the Y= editor, the wizard will paste the function into the equation.
2ndDISTR7
The first field is the x value: press X,T,θ,n to enter X. Fill in the other fields and press ▼︎ after each.
X,T,θ,n▼︎9▼︎
Press ENTER on Paste to place the function into Y₁.
ENTER
Press GRAPH to draw the curve. For the function χ²pdf(X,9), Y₁(7) = 0.1041197748.
GRAPH

χ²cdf(p. 245

χ²cdf(

χ²cdf(lowerbound,upperbound,df)

Probability that a χ² variable with df degrees of freedom (an integer > 0) lies between lowerbound and upperbound.

Keys: 2ndDISTR8

χ²cdf(0,3,4) → .4421745996

Note To mean infinity, type 1E99 (12ndEE99) for an upper bound, or ⁻1E99 for a lower bound. For example, to see the area to the left of an upper bound, make the lower bound ⁻1E99.

Tutorial

Evaluate χ²cdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 8:χ²cdf( by pressing 8. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR8
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
0▼︎3▼︎4▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate χ²cdf(0,3,4). The result is .4421745996.
ENTER

Fpdf(p. 245

Fpdf(

Fpdf(x,numerator df,denominator df)

Probability density function of the F distribution at x ≥ 0. Both degrees of freedom must be integers > 0. Paste it into Y= to plot the curve.

Keys: 2ndDISTR9

Fpdf(1,3,5) → .3611744789

Note A good window for plotting: Xmin=0, Xmax=5, Ymin=0, Ymax=1.

Tutorial

Evaluate Fpdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 9:Fpdf( by pressing 9. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR9
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
1▼︎3▼︎5▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate Fpdf(1,3,5). The result is .3611744789.
ENTER

Tutorial

Plot an F curve in Y=

Graph the F density with 24 and 19 degrees of freedom.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Set the window so the whole curve fits: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW0ENTER5ENTER1ENTER0ENTER1ENTER.1ENTER
Open the Y= editor with the cursor on Y₁.
Y=
Press 2ndDISTR and choose 9:Fpdf( by pressing 9. Because you are in the Y= editor, the wizard will paste the function into the equation.
2ndDISTR9
The first field is the x value: press X,T,θ,n to enter X. Fill in the other fields and press ▼︎ after each.
X,T,θ,n▼︎24▼︎19▼︎
Press ENTER on Paste to place the function into Y₁.
ENTER
Press GRAPH to draw the curve. For the function Fpdf(X,24,19), Y₁(1) = 0.9078268366.
GRAPH

Fcdf(p. 246

Fcdf(

Fcdf(lowerbound,upperbound,numerator df,denominator df)

Probability that an F variable lies between lowerbound and upperbound. Both degrees of freedom must be integers > 0.

Keys: 2ndDISTR0

Fcdf(0,2,3,5) → .767376082

Note To mean infinity, type 1E99 (12ndEE99) for an upper bound, or ⁻1E99 for a lower bound. For example, to see the area to the left of an upper bound, make the lower bound ⁻1E99.

Tutorial

Evaluate Fcdf(

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose 0:Fcdf( by pressing 0. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTR0
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
0▼︎2▼︎3▼︎5▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate Fcdf(0,2,3,5). The result is .767376082.
ENTER

binompdf(p. 246

binompdf(

binompdf(numtrials,p[,x])

Probability of exactly x successes in numtrials independent trials, each with success probability p. x can be an integer or a list of integers; 0 ≤ p ≤ 1 and numtrials must be an integer > 0. If you leave x out, you get a list of the probabilities for 0 through numtrials successes. The formula is f(x)=C(n,x)·pˣ·(1−p)ⁿ⁻ˣ.

Keys: 2ndDISTRALPHAA

binompdf(10,.5,5) → .24609375
binompdf(5,.3,{0,1,2}) → {.16807 .36015 .3087}

Tutorial

Exactly 5 heads in 10 tosses

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose A:binompdf( by pressing ALPHAA. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAA
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
10▼︎.5▼︎5▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate binompdf(10,.5,5). The result is .24609375.
ENTER

binomcdf(p. 247

binomcdf(

binomcdf(numtrials,p[,x])

Cumulative probability of at most x successes in numtrials trials with success probability p. x can be a real number or a list. If x is left out, you get a list of the cumulative probabilities for 0 through numtrials successes. 0 ≤ p ≤ 1 and numtrials must be an integer > 0.

Keys: 2ndDISTRALPHAB

binomcdf(10,.5,5) → .623046875

Tutorial

At most 5 heads in 10 tosses

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose B:binomcdf( by pressing ALPHAB. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAB
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
10▼︎.5▼︎5▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate binomcdf(10,.5,5). The result is .623046875.
ENTER

poissonpdf(p. 247

poissonpdf(

poissonpdf(μ,x)

Probability of exactly x events for a Poisson distribution with mean μ (a real number > 0). x can be an integer or a list of integers. The formula is f(x)=e^(−μ)·μˣ/x!, for x=0,1,2,….

Keys: 2ndDISTRALPHAC

poissonpdf(3,2) → .2240418077

Tutorial

Exactly 2 events when the mean is 3

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose C:poissonpdf( by pressing ALPHAC. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAC
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
3▼︎2▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate poissonpdf(3,2). The result is .2240418077.
ENTER

poissoncdf(p. 247

poissoncdf(

poissoncdf(μ,x)

Cumulative probability of at most x events for a Poisson distribution with mean μ (a real number > 0). x can be a real number or a list.

Keys: 2ndDISTRALPHAD

poissoncdf(3,2) → .4231900811

Tutorial

At most 2 events when the mean is 3

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose D:poissoncdf( by pressing ALPHAD. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAD
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
3▼︎2▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate poissoncdf(3,2). The result is .4231900811.
ENTER

geometpdf(p. 248

geometpdf(

geometpdf(p,x)

Probability that the first success happens on trial number x, for a geometric distribution with success probability p (0 ≤ p ≤ 1). x can be an integer or a list of integers. The formula is f(x)=p·(1−p)ˣ⁻¹, for x=1,2,….

Keys: 2ndDISTRALPHAE

geometpdf(.2,3) → .128

Tutorial

First success on the third trial

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose E:geometpdf( by pressing ALPHAE. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAE
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
.2▼︎3▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate geometpdf(.2,3). The result is .128.
ENTER

geometcdf(p. 248

geometcdf(

geometcdf(p,x)

Cumulative probability that the first success happens on or before trial x, for a geometric distribution with success probability p (0 ≤ p ≤ 1). x can be a real number or a list.

Keys: 2ndDISTRALPHAF

geometcdf(.2,3) → .488

Tutorial

First success within three trials

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

Press 2ndDISTR to open the DISTR menu and choose F:geometcdf( by pressing ALPHAF. With STAT WIZARDS on (the default) a small input screen appears.
2ndDISTRALPHAF
Fill in the fields. Type a value and press ▼︎ to move on; just press ▼︎ to keep a default.
.2▼︎3▼︎
The cursor is now on Paste. Press ENTER to paste the finished command onto the home screen.
ENTER
Press ENTER again to evaluate geometcdf(.2,3). The result is .488.
ENTER

Distribution Shadingp. 248

The DISTR DRAW instructions draw a density curve, shade the area between a lower and an upper bound, and print the area on the graph. Press 2ndDISTR ▶︎ to open the menu.

Choosing an item opens a wizard for its arguments. Optional arguments show their defaults; for a required argument the cursor will not move on until you enter a value. If you paste one of these instructions from the CATALOG instead, no wizard appears and you type the arguments yourself.

Note Set the window variables before drawing, so that the whole distribution fits on the screen. To erase a drawing, use 1:ClrDraw from the DRAW menu (2ndDRAW).

Note To mean infinity, type 1E99 (12ndEE99) for an upper bound, or ⁻1E99 for a lower bound. For example, to see the area to the left of an upper bound, make the lower bound ⁻1E99.

ShadeNorm(p. 249

ShadeNorm(

ShadeNorm(lowerbound,upperbound[,μ,σ])

Draws the normal density curve for mean μ and standard deviation σ and shades the area between lowerbound and upperbound. The defaults are μ=0 and σ=1.

Keys: 2ndDISTR▶︎1

ShadeNorm(65,1E99,63,2) shades the area above 65 under a normal curve with μ=63, σ=2.

Note A good window for the example above: Xmin=55, Xmax=72, Ymin=⁻.05, Ymax=.2.

Tutorial

Shade the upper tail of a normal curve

Shade the area above 65 when μ = 63 and σ = 2.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

The curve must fit on the screen, so set the window first: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW55ENTER72ENTER5ENTER(−).05ENTER.2ENTER0ENTER
Return to the home screen.
2ndQUIT
Press 2ndDISTR ▶︎ to open DISTR DRAW and choose 1:ShadeNorm( by pressing 1.
2ndDISTR▶︎1
Enter the arguments, pressing ▼︎ after each (for infinity, type 1 2ndEE 99, which is 1E99).
65▼︎12ndEE99▼︎63▼︎2▼︎
The cursor is on DRAW. Press ENTER to draw the curve and shade the area.
ENTER

Shade_t(p. 249

Shade_t(

Shade_t(lowerbound,upperbound,df)

Draws the Student-t density curve with df degrees of freedom and shades the area between lowerbound and upperbound.

Keys: 2ndDISTR▶︎2

Shade_t(⁻2,2,5)

Note A good window for the example above: Xmin=⁻3, Xmax=3, Ymin=⁻.15, Ymax=.5.

Tutorial

Shade the middle of a t curve

Shade the area between ⁻2 and 2 under the t curve with 5 degrees of freedom.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

The curve must fit on the screen, so set the window first: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW(−)3ENTER3ENTER1ENTER(−).15ENTER.5ENTER0ENTER
Return to the home screen.
2ndQUIT
Press 2ndDISTR ▶︎ to open DISTR DRAW and choose 2:Shade_t( by pressing 2.
2ndDISTR▶︎2
Enter the arguments, pressing ▼︎ after each (for infinity, type 1 2ndEE 99, which is 1E99).
(−)2▼︎2▼︎5▼︎
The cursor is on DRAW. Press ENTER to draw the curve and shade the area.
ENTER

Shadeχ²(p. 250

Shadeχ²(

Shadeχ²(lowerbound,upperbound,df)

Draws the χ² density curve with df degrees of freedom and shades the area between lowerbound and upperbound.

Keys: 2ndDISTR▶︎3

Shadeχ²(18.307,1E99,10)

Note A good window for the example above: Xmin=0, Xmax=35, Ymin=⁻.025, Ymax=.1.

Tutorial

Shade the upper tail of a χ² curve

Shade the area to the right of 18.307 under the χ² curve with 10 degrees of freedom (about 5 percent).

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

The curve must fit on the screen, so set the window first: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW0ENTER35ENTER5ENTER(−).025ENTER.1ENTER0ENTER
Return to the home screen.
2ndQUIT
Press 2ndDISTR ▶︎ to open DISTR DRAW and choose 3:Shadeχ²( by pressing 3.
2ndDISTR▶︎3
Enter the arguments, pressing ▼︎ after each (for infinity, type 1 2ndEE 99, which is 1E99).
18.307▼︎12ndEE99▼︎10▼︎
The cursor is on DRAW. Press ENTER to draw the curve and shade the area.
ENTER

ShadeF(p. 250

ShadeF(

ShadeF(lowerbound,upperbound,numerator df,denominator df)

Draws the F density curve for the given numerator and denominator degrees of freedom and shades the area between lowerbound and upperbound.

Keys: 2ndDISTR▶︎4

ShadeF(2.1,1E99,24,19)

Note A good window for the example above: Xmin=0, Xmax=5, Ymin=⁻.25, Ymax=1.

Tutorial

Shade the upper tail of an F curve

Shade the area to the right of 2.1 under the F curve with 24 and 19 degrees of freedom.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

The curve must fit on the screen, so set the window first: press WINDOW and enter Xmin, Xmax, Xscl, Ymin, Ymax and Yscl, pressing ENTER after each.
WINDOW0ENTER5ENTER1ENTER(−).25ENTER1ENTER0ENTER
Return to the home screen.
2ndQUIT
Press 2ndDISTR ▶︎ to open DISTR DRAW and choose 4:ShadeF( by pressing 4.
2ndDISTR▶︎4
Enter the arguments, pressing ▼︎ after each (for infinity, type 1 2ndEE 99, which is 1E99).
2.1▼︎12ndEE99▼︎24▼︎19▼︎
The cursor is on DRAW. Press ENTER to draw the curve and shade the area.
ENTER