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Chapter 10: Problem 34

The article "Theaters Losing Out to Living Rooms" (San Luis Obispo Tribune, June 17,2005 ) states that movie attendance declined in 2005 . The Associated Press found that 730 of 1000 randomly selected adult Americans preferred to watch movies at home rather than at a movie theater. Is there convincing evidence that the majority of adult Americans prefer to watch movies at home? Test the relevant hypotheses using a \(.05\) significance level.

Short Answer

Expert verified
Yes, there is convincing evidence that the majority of adult Americans prefer to watch movies at home. The test statistic (z = 14.56), associated with a virtually zero p-value, supports claiming that the proportion of adults who prefer to watch movies at home is greater than 0.5, making us reject the null hypothesis at a 0.05 significance level.

Step by step solution

01

State the hypotheses.

The null hypothesis (H0) will be that the proportion of American adults who prefer to watch movies at home is less than or equal to 0.5. This can be written as: H0: p <= 0.5. The alternative hypothesis (H1) will be that the proportion of American adults who prefer to watch movies at home is greater than 0.5. This can be written as: H1: p > 0.5.
02

Compute the test statistic.

We can use the z-score formula for testing a population proportion. First, find the sample proportion which is 730 / 1000 = 0.73. Then subtract the hypothesized population proportion (0.5) from the sample proportion, and divide by the standard deviation of the distribution of the sample proportion. The standard deviation (sd) is calculated as: sd = sqrt[(0.5*(1-0.5))/1000] = 0.0158. The z-score then is calculated as: z = (0.73-0.5)/0.0158 = 14.56.
03

Determine the p-value.

The p-value is the probability that you would observe a more extreme test statistic in the direction of the alternative hypothesis, assuming the null hypothesis were true. Because the test is one-tailed (as we are looking for evidence of a proportion greater than 0.5), we find the area to the right of our test statistic on the z-distribution. The z-score calculated is 14.56. This value is quite extreme, thus, the corresponding p-value is practically zero.
04

Make a decision.

Since the p-value is less than our significance level of 0.05, we reject the null hypothesis. This means we have strong evidence to believe that the majority of American adults prefer to watch movies at home.

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Most popular questions from this chapter

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: 379-390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

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