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Chapter 12

Statistical Inference: Other One-Sample Test Statistics

I One-Sample z Test for a Population Proportion, p

A. Introduction to z Test for a Population Proportion

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1. The binomial function rule

can be used to determine the probability of r

successes in n independent trials.

2. When n is large, the normal distribution can be

used to approximate the probability of r or more

successes. The approximation is excellent if

(a) the population is at least 10 times larger than

the sample and (b) np0 > 15 and n(1 – p0) > 15,

where p0 is the hypothesized proportion.

p( X r) nCr pr ( p 1)n r

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B. z Test Statistic for a Proportion

= sample estimator of the population

proportion

p0 = hypothesized population proportion

n = size of the sample used to compute

number of successes in the random sample

number of observations in the random sample

p̂

p̂

1. npp

ppz

00

0

1

ˆ

4

H0 : p p0

H0 : p p0 H0 : p p0

H1 : p p0

H1 : p p0 H1 : p p0

C. Statistical Hypotheses for a Proportion

population standard error of a proportion,

p p(1 p / n,

where p denotes the population proportion.

2. is an estimator of the nppp 00 1ˆ

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D. Computational Example

1. Student Congress believes that the proportion of

parking tickets issued by the campus police this

year is greater than last year. Last year the

proportion was p0 = .21.

2. To test the hypotheses

they obtained a random sample of n = 200

students and found that the proportion who

received tickets this year was

H0 : p .21

H1 : p .21

.27.ˆ p

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z.05 = 1.645

3. The null hypothesis can be rejected; the campus

police are issuing more tickets this year.

08.2

20021.121.

21.27.

1

ˆ

00

0

npp

ppz

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E. Assumptions of the z Test for a Population Proportion

1. Random sampling from the population

2. Binomial population

3. np0 > 15 and n(1 – p0) > 15

4. The population is at least 10 times larger than the

sample

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II One-Sample Confidence Interval for a Population Proportion, p

A. Two-Sided Confidence Interval

1. p̂ z /2p̂(1 p̂)

n p p̂ z /2

p̂(1 p̂)

n

population standard error of a proportion.

nppp ˆ1ˆˆ 2. is an estimator of the

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B. One-Sided Confidence Interval

1. Lower confidence interval

2. Upper confidence interval

p̂ zp̂(1 p̂)

n p

p p̂ zp̂(1 p̂)

n

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C. Computational Example Using the Parking Ticket Data

1. Two-sided 100(1 – .05)% = 95% confidence

interval

p̂ z /2p̂(1 p̂)

n p p̂ z /2

p̂(1 p̂)

n

.27 1.96

.27(1 .27)

200 p .27 1.96

.27(1 .27)

200

.208 p .332

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2. One-sided 100(1 – .05)% = 95% confidence

interval

p̂ zp̂(1 p̂)

n p

.27 1.645

.27(1 .27)

200 p

.218 p

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3. Comparison of the one- and two-sided confidence

intervals

Two-sided interval

One-sided interval

.30

L2 = .332 L1 = .208

.35.25.20p

.30

L1 = .218

.35.25.20p

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D. Assumptions of the Confidence Interval for a Population Proportion

1. Random sampling from the population

2. Binomial population

3. np0 > 15 and n(1 – p0) > 15

4. The population is at least 10 times larger than the

sample

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III Selecting a Sample Size, n

A. Information needed to specify n

1. Acceptable margin of error, m*, in

estimating p. m* is usually between .02

and .04.

2. Acceptable confidence level: usually .95

for z.05 or z.05/2

3. Educated guess, denoted by p*, of the

likely value of p

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B. Computational Example for the Traffic Ticket Data

1. One-sided confidence interval, let m* = .04,

z.05 = 1.645, and p* = .27

n z.05

m *

2

p * (1 p*)

n

1.645

.04

2

.27(1 .27) 333

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C. Conservative Estimate of the Required Sample Size

1. If a researcher is unable to provide an educated

guess for m*, a conservative estimate of n is

obtained by letting p* = .50.

n z.05

m *

2

p * (1 p*)

n

1.645

.04

2

.50(1 .50) 423

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IV One-Sample t Test for Pearson’s Population Correlation

A. t Test for 0 = 0 (Population Correlation Is Equal to Zero)

1. Values of | r | that lead to rejecting one of the

following null hypotheses are obtained from

Appendix Table D.6.

H0 : 0

H1 : 0

H0 : 0

H1 : 0

H0 : 0

H1 : 0

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Appendix Table D.6. Critical Values of the Pearson r

Degrees of

Freedom

n 2

Level of Significance for a One-Tailed Test

Level of Significance for a Two-Tailed Test

.05

.05

.025

.10

.01

.02 .01

.005

8 0.549 0.632 0.716 0.765

10 0.497 0.576 0.658 0.708

20 0.360 0.423 0.492 0.537

30 0.296 0.349 0.409 0.449

60 0.211 0.250 0.274 0.325

100 0.164 0.195 0.230 0.254

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1. Table D.6 is based on the t distribution and t

statistic

B. Computational Example Using the Girl’s Basketball Team Data (Chapter 5)

1. r = .84, n = 10, and r.05, 8 = .549

2. r.05, 8 = .549 is the one-tailed critical value from

Appendix Table D.6.

H0 : 0,

freedom of degrees 2 with 1

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n

r

nrt

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1. Because r = .84 > r.05, 8 = .549, reject the null

hypothesis and conclude that player’s height and

weight are positively correlated.

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C. Assumptions of the t Test for Pearson’s Population Correlation Coefficient

1. Random sampling

2. Population distributions of X and Y are

approximately normal.

3. The relationship between X and Y is linear.

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4. The distribution of Y for any value of X is

normal with variance that does not depend on the

X value selected and vice versa.

V One-Sample Confidence Interval for Pearson’s Population Correlation

A. Fisher’s r to Z Transformation

1. r is bounded by –1 and +1; Fisher’s Z can

exceed –1 and +1.

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Appendix Table D.7 Transformation of r to Z

0.200 0.203 0.400 0.424 0.600 0.693 0.800 1.099

0.225 0.229 0.425 0.454 0.625 0.733 0.825 1.172

0.250 0.255 0.450 0.485 0.650 0.775 0.850 1.256

0.275 0.282 0.475 0.517 0.675 0.820 0.875 1.354

0.300 0.310 0.500 0.549 0.700 0.867 0.900 1.472

0.325 0.337 0.525 0.583 0.725 0.918 0.925 1.623

0.350 0.365 0.550 0.618 0.750 0.973 0.950 1.832

0.375 0.394 0.575 0.655 0.775 1.033 0.975 2.185

r r r r Z Z Z Z

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B. Two Sided Confidence Interval for Using Fisher’s Z Transformation

1. Begin by transforming r to Z. Then obtain

a confidence interval for ZPop

Z z.05/2

1

n 3 ZPop Z z.05/2

1

n 3

2. A confidence interval for r is obtained by

transforming the lower and upper confidence

limits for ZPop into r using Appendix Table D.6 .

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C. One-Sided Confidence Interval for

1. Lower confidence limit

Z z.05

1

n 3 ZPop

2. Upper confidence limit

ZPop Z z.05

1

n 3

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D. Computational Example Using the Girl’s Basketball Team Data (Chapter 5)

1. r = .84, n = 10, and Z = 1.221

Z z.05

1

n 3 ZPop

1.221 1.645

1

10 3 ZPop

.599 ZPop

.54

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3. A confidence interval can be used to test

hypotheses for any hypothesized value of 0.

For example, any hypothesis for which 0 ≤ .54

could be rejected.

.60

L1 = .54

.65.55.50

2. Graph of the confidence interval for

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E. Assumptions of the Confidence Interval for Pearson’s Correlation Coefficient

1. Random sampling

2. is not too close to 1 or –1

3. Population distributions of X and Y are

approximately normal

4. The relationship between X and Y is linear

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5. The distribution of Y for any value of X is

normal with variance that does not depend on the

X value selected and vice versa.

VI Practical Significance of Pearson’s Correlation

A. Cohen’s Guidelines for Effect Size

r = .10 is a small strength of association

r = .30 is a medium strength of association

r = .50 is a large strength of association