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Chapter 10: Q 14. (page 640)

Artificial trees? An association of Christmas tree growers in Indiana wants to know if there is a difference in preference for natural trees between urban and rural households. So the association sponsored a survey of Indiana households that had a Christmas tree last year to find out. In a random sample of 160rural households, 64had a natural tree. In a separate random sample of 261urban households, 89had a natural tree. A 95%confidence interval for the difference (Rural – Urban) in the true proportion of households in each population that had a natural tree is -0.036to0.154. Does the confidence interval provide convincing evidence that the two population proportions are equal? Explain your answer.

Short Answer

Expert verified

It provides convincing evidence that two population proportion are equal.

Step by step solution

01

Given Information

It is given that we have(-0.036,0.154)at95%confidence interval.

02

Explanation

The interval (-0.036,0.154)contains zero, It is likely that difference of proportion is zero. Two population proportions can be equal.

It implies that there is possibility that population proportions are equal.

It can be concluded that there is no convincing evidence that two population proportions are not equal.

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Most popular questions from this chapter

A random sample of size n will be selected from a population, and the proportion p^3051526=0.200=20.0%p^ of those in the sample who have a Facebook page will be calculated. How would the margin of error for a 95% confidence interval be affected if the sample size were increased from 50to200 and the sample proportion of people who have a Facebook page is unchanged?

a. It remains the same.

b. It is multiplied by 2.

c. It is multiplied by 4.

d. It is divided by 2.

e. It is divided by 4.

Researchers want to evaluate the effect of a natural product on reducing blood pressure. They plan to carry out a randomized experiment to compare the mean reduction in blood pressure of a treatment (natural product) group and a placebo group. Then they will use the data to perform a test of H0:μT−μP=0versus Ha:μT−μP>0, where μT= the true mean reduction in blood pressure when taking the natural product and μP = the true mean reduction in blood pressure when taking a placebo for subjects like the ones in the experiment. The researchers would like to detect whether the natural product reduces blood pressure by at least 7points more, on average, than the placebo. If groups of size 50are used in the experiment, a twosample t test using α=0.01will have a power of 80%to detect a 7-point difference in mean blood pressure reduction. If the researchers want to be able to detect a 5-point difference instead, then the power of the test

a. would be less than 80%.

b. would be greater than 80%.

c. would still be 80%.

d. could be either less than or greater than 80%.

e. would vary depending on the standard deviation of the data

Two samples or paired data? In each of the following settings, decide whether you should use two-sample t procedures to perform inference about a difference in means or paired t procedures to perform inference about a mean difference. Explain your choice.

a. To compare the average weight gain of pigs fed two different diets, nine pairs of pigs were used. The pigs in each pair were littermates. A coin toss was used to decide which pig in each pair got Diet A and which got Diet B.

b. Separate random samples of male and female college professors are taken. We wish to compare the average salaries of male and female teachers.

c. To test the effects of a new fertilizer, 100 plots are treated with the new fertilizer, and 100 plots are treated with another fertilizer. A computer’s random number generator is used to determine which plots get which fertilizer.

Mrs. Woods and Mrs. Bryan are avid vegetable gardeners. They use different fertilizers, and each claims that hers is the best fertilizer to use when growing tomatoes. Both agree to do a study using the weight of their tomatoes as the response variable. Each planted the same varieties of tomatoes on the same day and fertilized the plants on the same schedule throughout the growing season. At harvest time, each randomly selects 15 tomatoes from her garden and weighs them. After performing a two-sample t test on the difference in mean weights of tomatoes, they gett=5.24 andP=0.0008. Can the gardener with the larger mean claim that her fertilizer caused her tomatoes to be heavier?

a. Yes, because a different fertilizer was used on each garden.

b. Yes, because random samples were taken from each garden.

c. Yes, because the P-value is so small.

d. No, because the condition of the soil in the two gardens is a potential confounding variable.

e. No, because15<30

American-made cars Nathan and Kyle both work for the Department of Motor Vehicles (DMV), but they live in different states. In Nathan’s state, 80%of the registered cars are made by American manufacturers. In Kyle’s state, only 60%of the registered cars are made by American manufacturers. Nathan selects a random sample of 100cars in his state and Kyle selects a random sample of 70cars in his state. Let pn-pkbe the difference (Nathan’s state – Kyle’s state) in the sample proportion of cars made by American manufacturers.

a. What is the shape of the sampling distribution of pn-pk? Why?

b. Find the mean of the sampling distribution.

c. Calculate and interpret the standard deviation of the sampling distribution.

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