Inference for Proportions

Moore IPS Chapter 8

© 2012 W.H. Freeman and Company

The correct rule for proportions is that we can create confidence intervals for p

based on:

a. if n is at least 10.

b. at least 15 successes and 15 failures.

c. if n ≥ 30.

8.1 Inference for a Single Proportion

8.1-1

In a sample of 20 items, I found six to be defective. In constructing a confidence

interval for the proportion of defectives, I should use:

a. the plus four method.

b. the large sample interval.

c. neither method.

8.1 Inference for a Single Proportion

8.1-2

The sample size is larger than 10, but fewer than 10 defectives, so the plus four method will be best.

A sample of 75 students found that 55 of them had cell phones. The

margin of error for a 95% confidence interval estimate for the

proportion of all students with cell phones is:

a. 0.084.

b. (0.633, 0.833).

c. 0.100.

8.1 Inference for a Single Proportion

8.1-3

*

55 20ˆ ˆ 75 751.96 0.100

75

pqz

n

It is known that 12% of all students taking a particular course receive a

grade of A. There are 155 students in one section of the course. Is the

use of the normal approximation justified in this case?

a. Yes, because np > 10 and nq > 10

b. Yes, because np >10 and npq > 10

c. No, because np > 10 and nq > 10

d. No, because np > 10 and npq > 10

8.1 Inference for a Single Proportion

8.1-4

It is known that 12% of all students taking a particular course receive a

grade of A. There are 155 students in one section of the course. What

is the probability that at least 14% of them receive a grade of A?

a. 0.7794

b. 0.1103

c. 0.2206

8.1 Inference for a Single Proportion

8.1-5

z 0.14 0.12

0.12(1 0.12)

155

0.77

P(Z 0.77) 0.2206

You buy a package of 122 Smarties and 19 of them are red. What is a 95% confidence interval for the true proportion of red Smarties?

a. (0.092, 0.220)

b. (0.103, 0.230)

c. (0.085, 0.199)

8.1 Inference for a Single Proportion

8.1-6

ˆ p z* ˆ p (1 ˆ p )

nˆ p 19 /122

We want to construct a 95% confidence interval for the true proportion of all adult males who have spent time in prison, with a margin of error of 0.02. From previous studies, we believe the proportion to be somewhere around 0.07. The required sample size is therefore:

a. 620.

b. 626.

c. 632.

8.1 Inference for a Single Proportion

8.1-7

n (z*

m)2( p*)(1 p*)

n 9604(0.07)(0.93)

In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast?

We will test Ho: p = 0.50 Ha: p > 0.50

What is the value of the test statistic?

a. 2.11

b. 5.62

c. 2.31

8.1 Inference for a Single Proportion

8.1-8

z 0.55 0.50

0.50(0.50)

446

2.11

In a sample of 446 students, 246 ate breakfast. Can we can conclude that more than 50% of all students eat breakfast?

We will test Ho: p = 0.50 Ha: p > 0.50

What is the p-value of the test?

a. 0.0174

b. 0.1765

c. 0.2876

8.1 Inference for a Single Proportion

8.1-9

p(z 2.11) 1 0.9826 0.0174

An article about a new drug stated that “the incidence of side

effects was similar to placebo, P-value > 0.05.”

With the information given:

a. one should reject the null hypothesis of no difference at 10%.

b. one should not reject the null hypothesis of no difference at 10%.

c. There is not enough information given.

8.1 Inference for a Single Proportion

8.1-10

We know the P-value is more than 5%, but we don't know how much more.

Without the actual P-value for the test, we cannot make a determination.

A student believes that 20% of all students think pepperoni is their favorite pizza.

He performs a test of hypothesis, H0: p = 0.2, having taken a sample of 200

students and finding that 52 think pepperoni is their favorite. He finds a P-value

of 0.0338, so rejects the null at = 0.05.

He then computes a 95% confidence interval for the true proportion and finds it is

(0.199, 0.321). He is confused. 20% is in the interval!

What is the difference?

a. He made a mistake in one of his calculations.

b. The two use different values of p in computing the standard deviation.

c. He should have used the plus four confidence interval.

8.1 Inference for a Single Proportion

8.1-11

The hypothesis test uses p0 = 0.2 in computing the standard deviation while the confidence interval uses p-hat = 0.26 in computing the standard error.

Drug sniffing dogs must be 95% accurate. A new dog is being tested and

is right in 46 of 50 trials. Find a 95% confidence interval for the

proportion of times the dog will be correct.

a. (0.845, 0.995)

b. (0.805, 0.973)

c. (0.819, 0.959)

8.1 Inference for a Single Proportion

8.1-12

use the plus four estimate … 2

4

Xp

n

Drug sniffing dogs must be 95% accurate. A new dog is being tested and

is right in 46 of 50 trials. Find a 95% confidence interval for the

proportion of times the dog will be correct.

a. (0.845, 0.995)

b. (0.805, 0.973)

c. (0.819, 0.959)

8.1 Inference for a Single Proportion

8.1-12c

use the plus four estimate … 2

4

Xp

n

C

A poll finds that 54% of the 600 people polled favor the incumbent.

Shortly after the poll is taken, it is disclosed that he had an extramarital

affair. A new poll finds that 50% of the 1030 polled now favor the

incumbent.

The standard error for a confidence interval for the candidate's latest

support level is:

a. 0.016.

b. 0.020.

c. 0.025.

8.1 Inference for a Single Proportion

8.1-13

ˆ ˆ(1 )ˆ( )

p pSE p

n

A noted psychic was tested for ESP. The psychic was presented with400 cards face down and asked to determine if each card was markedwith one of four different symbols. The psychic was correct in 120cases. Let p represent the probability that the psychic correctly identifiesthe symbol on the card in a random trial. Suppose you wish to see ifthere is evidence that the psychic was doing better than just guessing.

To do this you test the hypotheses:

H0: p = 0.25,

Ha: p > 0.25.

The P-value of your test is:

a. 0.0104.b. 0.0146.c. 0.9896.

8.1 Inference for a Single Proportion

8.1-14

C

Suppose you want to know which of two manufacturing methods will bebetter. You create 10 prototypes using the first process, and 10 usingthe second. There were 3 defectives in the first batch and 5 in thesecond.

A 90% confidence interval for the difference in the two proportions is (-0.493, 0.159).

What conclusion should I make about the two manufacturing processes?

a. The first method is better.

b. The second method is better.

c. Both methods may be equivalent.

8.2 Comparing Two Proportions

8.2-1

You want to know which of two manufacturing methods will be better.

You create 10 prototypes using the first process, and 10 using the

second. There were 3 defectives in the first batch and 5 in the second.

Find a 95% confidence interval for the difference in the proportion

of defectives.

a. (-0.62, 0.22)

b. (-0.56, 0.22)

c. (-0.493, 0.160)

8.2 Comparing Two Proportions

8.2-2

The numbers of trials and defectives are small, so we should use the plus four estimates, where we add 2 trials to each sample and 1 defective to each sample.

You want to know which of two manufacturing methods will be better.

You create 10 prototypes using the first process, and 10 using the

second. There were 3 defectives in the first batch and 5 in the second.

Find a 95% confidence interval for the difference in the proportion

of defectives.

a. (-0.62, 0.22)

b. (-0.56, 0.22)

c. (-0.493, 0.160)

8.2 Comparing Two Proportions

8.2-2c

The numbers of trials and defectives are small, so we should use the plus four estimates, where we add 2 trials to each sample and 1 defective to each sample.

C

A poll finds that 54% of the 600 people polled favor the incumbent.

Shortly after the poll is taken, it is disclosed that he had an extramarital

affair. A new poll finds that 50% of the 1030 polled now favor the

incumbent. We want to know if his support has decreased. In computing

a test of hypothesis with , what is the estimate of the overall

proportion?

a. 52%

b. 52.5%

c. 51.5%

X1 = .54*600 = 324 X2 = 0.5*1030 = 515 So, total of 600 + 1030 = 1630. Hence:

8.2 Comparing Two Proportions

8.2-3

0 1 2:H p p

324 515ˆ .515

1630p

A poll finds that 54% of the 600 people polled favor the incumbent.

Shortly after the poll is taken, it is disclosed that he had an extramarital

affair. A new poll finds that 50% of the 1030 polled now favor the

incumbent.

We want to know if his support has decreased. The test statistic is

a. z = 1.56

b. z = -2.57

c. z = -1.55

8.2 Comparing Two Proportions

8.2-4

.

.54 .50

1 1.515 (1 .515)

600 1030

z

324 515ˆ .515

1630p

A poll finds that 54% of the 600 people polled favor the incumbent.

Shortly after the poll is taken, it is disclosed that he had an extramarital

affair. A new poll finds that 50% of the 1030 polled now favor the

incumbent.

We want to know if his support has decreased. The test statistic is

a. z = 1.56

b. z = -2.57

c. z = -1.55

8.2 Comparing Two Proportions

8.2-4c

.

C

Based on surveys conducted in 1989 and 1999, a researcher compared the

proportion of high school age females interested in a career in science in 1989

with the proportion in 1999.

He concluded that the proportions were not significantly different at the = 0.05

level because the P-value was 0.121. Assuming the surveys were simple

random samples from the appropriate populations, we may conclude:

a. that the probability of observing a difference at least as large as that

observed by the researcher if, in fact, the two proportions were equal is

0.121.

b. that in repeated sampling, the researcher would obtain the difference actually observed in approximately 12.1% of the samples.

c. very little. Without knowing if the observed difference is practically significant, we cannot assess whether the results are statistically significant.

8.2 Comparing Two Proportions

8.2-5

An SRS of 100 of a certain popular model car in 1993 found that 20

had a certain minor defect in the brakes. An SRS of 400 of this model

car in 1994 found that 50 had the minor defect in the brakes. Let p1 and

p2 be the proportion of all cars of this model in 1993 and 1994,

respectively, that actually contain the defect.

A 90% confidence interval for p1 – p2 is:

a. 0.075 ± 0.084.

b. 0.075 ± 0.071.

c. 0.075 ± 0.043.

8.2 Comparing Two Proportions

8.2-6c

C

A manufacturer receives parts from two suppliers. An SRS of 400 parts

from supplier 1 finds 20 defective. An SRS of 100 parts from supplier 2

finds 10 defective. Let p1 and p2 be the proportion of all parts from

suppliers 1 and 2, respectively, that are defective. Is there evidence of

a difference in the proportion of defective parts produced by the two

suppliers? To determine this, you test the hypotheses:

H0: p1 = p2 Ha: p1 ≠ p2.

The P -value of your test is:

a. 0.1164.

b. 0.060.

c. 0.0301.

8.2 Comparing Two Proportions

8.2-7c

C

A psychologist claims she has developed a cognitive therapy program that is more effective in helping smokers quit smoking than anything currently available. In particular, she claims that her program is more effective than the nicotine patch, which is widely used by smokers trying to quit.

A sample of 75 adult smokers who have indicated a desire to quit is located. The subjects are randomized into two groups. The cognitive therapy program will be administered to the 38 smokers in the first group, and the 37 smokers in the second group will use the nicotine patch. After a period of one year, each subject indicates whether he or she has successfully quit smoking.

In the therapy group, 22 people say they have quit smoking, while 17 people who used the patch have quit. What is the value of the test statistic for this claim?

a. 1.04

b. 1.14

c. 1.24

8.2 Comparing Two Proportions

8.2-8

z ˆ p 1 ˆ p 2

ˆ p (1 ˆ p )1

n1

1

n2

0.579 0.459

0.520(0.480)138

137

0.120

0.0133141.04