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Classic Graphing 84 manual · Chapter 17 of 25

Hypothesis tests and confidence intervals

Z, t, proportion, chi-square, F, regression and ANOVA tests, and the matching confidence intervals, with Data or Stats input.

17.1The TESTS menu#

Press stat then ◀ for the TESTS tab (or ▶ twice). Each item opens a form. Choose Data (use a list) or Stats (type the summary numbers) with Inpt, fill in the fields moving with ▼, and finish on Calculate (or Draw, which also shades the p-value area on a graph).

The TESTS tab. The display shows: EDIT CALC [TESTS] / 1:Z-Test… / 2:T-Test… / 3:2-SampZTest… / 4:2-SampTTest… / 5:1-PropZTest… / 6:2-PropZTest… / 7↓ZInterval….
The TESTS tab.
ItemTest or interval
1 Z-TestMean of one sample, σ known
2 T-TestMean of one sample, σ unknown
3 2-SampZTestTwo means, σ known
4 2-SampTTestTwo means, σ unknown (pooled or not)
5 1-PropZTestOne proportion
6 2-PropZTestTwo proportions
7 ZIntervalConfidence interval for a mean, σ known
8 TIntervalConfidence interval for a mean, σ unknown
9 2-SampZIntInterval for a difference of means, σ known
0 2-SampTIntInterval for a difference of means, σ unknown
A 1-PropZIntInterval for a proportion
B 2-PropZIntInterval for a difference of proportions
C χ²-TestChi-square test on a matrix (default [A], expected goes to [B])
D χ²GOF-TestGoodness of fit with Observed and Expected lists and df
E 2-SampFTestCompare two standard deviations
F LinRegTTestTest the slope of a regression line
G LinRegTIntConfidence interval for the slope
H ANOVA(One-way analysis of variance: ANOVA(L1,L2,…)

The test alternative is chosen on its own row, for example μ: ≠μ0, <μ0 or >μ0. Results are also stored in the statistics variables (p, t, z, df and so on).

17.2Test a mean (T-Test)#

t=xˉ−μ0s/n,df=n−1t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}},\qquad df=n-1
The T-Test form with Data input. The display shows: T-Test / Inpt:[Data] Stats / μ0:0 / List:L1 / Freq:1 / μ:[≠μ0] <μ0 >μ0 / Calculate Draw.
The T-Test form with Data input.
Example 17.1

T-Test from summary numbers

t=5.5−51.2/30t=\frac{5.5-5}{1.2/\sqrt{30}}

  1. Open T-Test, move right to Stats and press enter. Fill μ0 = 5, x̄ = 5.5, Sx = 1.2, n = 30.

    stat◀2▶enter▼5▼5.5▼1.2▼30

  2. Move down to Calculate and press enter.

    ▼▼enter
Result of the example "T-Test from summary numbers" The display shows: T-Test / μ≠5 / t=2.282177323 / p=.0299998166 / df=29 / x̄=5.5 / Sx=1.2 / n=30.
You should seeμ≠5 t=2.282177323 p=.0299998166 df=29
Example 17.2

T-Test on a list, one-sided

  1. L1 = 1 to 5. Open T-Test (Data), μ0 = 2, alternative >μ0.

    stat11enter2enter3enter4enter5enterstat◀2▼2▼▼▼▶▶enter▼enter
Result of the example "T-Test on a list, one-sided" The display shows: T-Test / μ>2 / t=1.414213562 / p=.1150998205 / df=4 / x̄=3 / Sx=1.58113883 / n=5.
You should seeμ>2 t=1.414213562 p=.1150998205

17.3Test a mean when σ is known (Z-Test)#

z=xˉ−μ0σ/nz=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}
Example 17.3

Z-Test from summary numbers

z=105−10015/36=2z=\frac{105-100}{15/\sqrt{36}}=2

  1. Choose Stats; μ0 = 100, σ = 15, x̄ = 105, n = 36; then Calculate.

    stat◀1▶enter▼100▼15▼105▼36▼▼enter
Result of the example "Z-Test from summary numbers" The display shows: Z-Test / μ≠100 / z=2 / p=.0455002639 / x̄=105 / n=36.
You should seeμ≠100 z=2 p=.0455002639
Example 17.4

Draw the p-value

  1. Same data, but move right to Draw at the bottom and press enter.

    stat◀1▶enter▼100▼15▼105▼36▼▼▶enter
Result of the example "Draw the p-value" The display shows: z=2 p=.04550026.
You should seez=2 p=.04550026

17.4Test a proportion (1-PropZTest)#

z=p^−p0p0(1−p0)/nz=\frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}
Example 17.5

60 successes in 100 trials against p0 = 0.5

  1. p0 = .5, x = 60, n = 100, then Calculate.

    stat◀5.5▼60▼100▼▼enter
Result of the example "60 successes in 100 trials against p0 = 0.5" The display shows: 1-PropZTest / prop≠.5 / z=2 / p=.0455002639 / p̂=.6 / n=100.
You should seeprop≠.5 z=2 p=.0455002639 p̂=.6

17.5Find a confidence interval#

xˉ±t∗sn\bar{x}\pm t^{*}\frac{s}{\sqrt{n}}
The interval for a mean when σ is unknown.
Example 17.6

ZInterval from summary numbers

  1. σ = 15, x̄ = 105, n = 36, C-Level .95 (the default).

    stat◀7▶enter▼15▼105▼36▼▼enter
Result of the example "ZInterval from summary numbers" The display shows: ZInterval / (100.10009,109.89991) / x̄=105 / n=36.
You should see(100.10009,109.89991) x̄=105 n=36
Example 17.7

TInterval from a list

  1. L1 = 1 to 5, then TInterval with Data: move down four times and press enter.

    stat11enter2enter3enter4enter5enterstat◀8▼▼▼▼enter
Result of the example "TInterval from a list" The display shows: TInterval / (1.036756839,4.963243161) / x̄=3 / df=4 / Sx=1.58113883 / n=5.
You should see(1.036756839,4.963243161) df=4

17.6Chi-square tests and ANOVA#

χ2=∑(O−E)2E\chi^2=\sum\frac{(O-E)^2}{E}
Example 17.8

Goodness of fit

  1. L1 = 10, 20, 30 (observed), L2 = 20, 20, 20 (expected). Press stat, left, alpha inv (D), move down twice, type 2 for df, down, enter.

    stat110enter20enter30enter▶20enter20enter20enterstat◀alphaDx⁻¹▼▼2▼enter
Result of the example "Goodness of fit" The display shows: χ²GOF-Test / χ²=10 / p=.006737947 / df=2 / CNTRB={5 0 5}.
You should seeχ²=10 p=.006737947 df=2 CNTRB={5 0 5}
Example 17.9

Chi-square test on a table

  1. Store [[10,20][30,40]] in [A], then choose χ²-Test (alpha prgm, C) and Calculate.

    2nd[×2nd[×10,202nd]−2nd[×30,402nd]−2nd]−sto→2ndMATRIXx⁻¹1enterstat◀alphaCprgm▼▼enter
Result of the example "Chi-square test on a table" The display shows: χ²-Test / χ²=.7936507937 / p=.3729984836 / df=1.
You should seeχ²-Test χ²=.7936507937
Example 17.10

One-way ANOVA

  1. Store {1,2,3} in L1 and {4,5,6} in L2. Choose ANOVA( (alpha pow, H), type L1,L2) and press enter.

    2nd{(1,2,32nd})sto→2ndL11enter2nd{(4,5,62nd})sto→2ndL22enterstat◀alphaH^2ndL11,2ndL22)enter
Result of the example "One-way ANOVA" The display shows: One-way ANOVA / F=13.5 / p=.0213116411 / Factor / df=1 / SS=13.5 / MS=13.5 / Error ↓.
You should seeOne-way ANOVA F=13.5
Example 17.11

A test result is kept in p

  1. Run the 1-PropZTest above and leave with clear.

    stat◀5.5▼60▼100▼▼enterclear

  2. Press vars, 5, left, left (TEST tab), 1 (p) and enter.

    vars5◀◀1enter
Result of the example "A test result is kept in p" The display shows: prop≠.5 / z=2 / p=.0455002639 / p̂=.6 / n=100 / p / .0455002639.
You should see.0455002639

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