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Chapter 8: Q68E (page 452)

Conducting a political poll. A pollster wants to estimate the difference between the proportions of men and women who favor a particular national candidate using a 90% confidence interval of width .04. Suppose the pollster has no prior information about the proportions. If equal numbers of men and women are to be polled, how large should the sample sizes be?

Short Answer

Expert verified

The required sample size is 3383.

Step by step solution

01

Given Information

With a 95% confidence interval, a pollster intends to calculate the actual difference between the proportion of men and women.

From the given information

2SE=0.04SE=0.02
02

Z-value

For 伪=0.1

The z-value is given by

z伪2=z0.05=1.645

03

Compute the sample

For, z=1.645,p1=p2=0.5鈥�andSE=0.02

The sample is calculated as

n1=n2=z0.052脳p11-p1+p21-p2SE2=1.6452脳0.25+0.250.022=2.706025脳.50.0004=1.35301250.0004=3382.53鈮�3383

Therefore, the required sample size is 3383.

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Most popular questions from this chapter

Ages of self-employed immigrants. Is self-employment for immigrant workers a faster route to economic advancement in the country? This was one of the questions studied in research published in the International Journal of Manpower (Vol. 32, 2011). One aspect of the study involved comparing the ages of self-employed and wage-earning immigrants. The researcher found that in Sweden, native wage earners tend to be younger than self-employed natives. However, immigrant wage earners tend to be older than self-employed immigrants. This inference was based on the table's summary statistics for male Swedish immigrants.

Self-employed immigrants

Wage-earning immigrants

Sample Size

870

84,875

Mean

44.88

46.79

Source: Based on L. Andersson, "Occupational Choice and Returns to Self-Employment Among Immigrants," International Journal of Manpower, Vol. 32, No. 8, 2011 (Table I).

a. Based on the information given, why is it impossible to provide a measure of reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden"?

b. What information do you need to measure reliability for the inference, part a?

c. Give a value of the test statistic that would conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants if you are willing to risk a Type I error rate of .01.

d. Assume that s, the standard deviation of the ages is the same for both self-employed and wage-earning immigrants. Give an estimate of s that would lead you to conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants using 伪=0.01 .

e. Is the true value of s likely to be larger or smaller than the one you calculated in part d?

To compare the means of two populations, independent random samples of 400 observations are selected from each population, with the following results:

Sample 1

Sample 2

x炉1=5,275蟽1=150

x炉2=5,240蟽2=200

a. Use a 95%confidence interval to estimate the difference between the population means (渭1鈭�渭2). Interpret the confidence interval.

b. Test the null hypothesis H0:(渭1鈭�渭2)=0versus the alternative hypothesis Ha:(渭1鈭�渭2)鈮�0 . Give the significance level of the test and interpret the result.

c. Suppose the test in part b was conducted with the alternative hypothesis Ha:(渭1鈭�渭2)鈮�0 . How would your answer to part b change?

d. Test the null hypothesis H0:(渭1鈭�渭2)=25 versus Ha:(渭1鈭�渭2)鈮�25. Give the significance level and interpret the result. Compare your answer with the test conducted in part b.

e. What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a鈥揹?

What is the line of means?

Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy x of a professional soccer player鈥檚 shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target,x=0; if he misses his target 2 feet to the right, x=2; and if he misses 1 foot to the left,x=-1.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where he is standing; all other shots on goal will score. Consider a goalkeeper who stands in the middle of the goal.

a. If the player aims for the right goalpost, what is the probability that he will score?

b. If the player aims for the center of the goal, what is the probability that he will score?

c. If the player aims for halfway between the right goal post and the outer limit of the goalkeeper鈥檚 reach, what is the probability that he will score?

Consider the discrete probability distribution shown here.

x

10

12

18

20

p

.2

.3

.1

.4

a. Calculate渭,蟽2 and蟽 .

b. What isP(x<15) ?

c. Calculate 渭卤2蟽 .

d. What is the probability that xis in the interval 渭卤2蟽 ?
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