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Chapter 11: Q.4 (page 480)

Fill in the blanks.

a. The mean of all possible sample proportions is equal to the

b. For large samples, the possible sample proportions have approximately a distribution.

c. A rule of thumb for using a normal distribution to approximate the distribution of all possible sample proportions is that both and are or greater.

Short Answer

Expert verified

a). The mean of sample proportions is equal to the population proportion.

b). For large samples, the possible sample proportions have approximately a normal distribution.

c). Hence, a rule of thumb for using a normal distribution to approximate of all possible sample proportions is that both np and n(1-p) are 5 or greater.

Step by step solution

01

Part (a) Step 1: Given Information

A symmetric probability distribution about the mean indicates that data close to the mean occur more frequently than those further away.

02

Part (a) Step 2: Explanation

A population proportion is a good characteristic of the population.

Hence, the mean of sample proportions is equal to the population proportion.

03

Part (b) Step 1: Given Information

A symmetric probability distribution about the mean indicates that data close to the mean occur more frequently than those further away.

04

Part (b) Step 2: Explanation

The normal distribution is the statistic's probability distribution.

Hence, for large samples, the possible sample proportions have approximately a normal distribution.

05

Part (c) Step 1: Given Information

A symmetric probability distribution about the mean, indicates that data close to the mean occur more frequently than those further away.

06

Part (c) Step 2: Explanation

The maximum value plus two standard deviations and the minimum value minus two standard deviations are the maximum and minimum values, respectively, in a normal distribution.

Hence, a rule of thumb for using a normal distribution to approximate of all possible sample proportions is that both np and n(1-p) are 5 or greater.

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Most popular questions from this chapter

a. Determine the sample proportion.

b. Decide whether using the one-proportion z-test is appropriate.

c. If appropriate, use the one-proportion z-test to perform the specified hypothesis test.

x=3

n=100

H0:p=0.04

Ha:p鈮�0.04

伪=0.10

Drinking Habits. In a nationwide survey, conducted by Pulse Opinion Research, LLC for Rasmussen Reports, 1000 American adults were asked, among other things, whether they drink alcoholic beverages at least once a week; 38% said "yes." Determine and interpret a 95% confidence interval for the proportion, p, of all American adults who drink alcoholic beverages at least once a week.

We have given a likely range for the observed value of a sample proportion p^

0.7orless

a. Based on the given range, identify the educated guess that should be used for the observed value of p^to calculate the required sample size for a prescribed confidence level and margin of error.

b. Identify the observed values of the sample proportion that will yield a larger margin of error than the one specified if the educated guess is used for the sample-size computation.

Explain the relationships among the sample proportion, the number of successes in the sample, and the sample size.

In discussing the sample size required for obtaining a confidence interval with a prescribed confidence level and margin of error, we made the following statement: "... we should be aware that, if the observed value of p^is closer to 0.5than is our educated guess, the margin of error will be larger than desired." Explain why.

One-Proportion Plus-Four z-Interval Procedure. To obtain a plus four z-interval for a population proportion, we first add two successes and two failures to our data (hence, the term "plus four") and then apply Procedure 11.1on page 454to the new data. In other words, in place of p^(which is x/n), we use p~=(x+2)/(n+4). Consequently, for a confidence level of 1-伪, the endpoints of the plus-four z-interval are

p~卤za/2路p~(1-p~)/(n+4)

As a rule of thumb, the one-proportion plus-four z-interval procedure should be used only with confidence levels of 90% or greater and sample sizes of 10 or more.
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