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In: Statistics and Probability

Audio Co. Manages a chain of stores that sell audio systems. It has been very successful,...

Audio Co. Manages a chain of stores that sell audio systems. It has been very successful, but also many failures. The analysis of these failures has led it to adopt the policy of not opening a store unless they are reasonably sure that at least 15% of the students of the place have stereo systems with a cost of $ 1,100 or more. In a survey of 300 of the 2,400 students at a small arts school in the Midwest, he found that 57 of them had a stereo system that cost more than or equal to $ 1,100. If your company wants to run a 5% failure risk, should you open a store in this place?

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Expert Solution

Research question: Do the sample data provides enough evidence to conclude that at least 15% of the students in the Midwest had a stereo system that cost more than or equal to $1,100.

Null hypothesis: H0: The population proportion of the students of Midwest having a stereo system that cost more than or equal to $1,100 is greater than or equal to 0.15.

Alternate hypothesis: Ha: The population proportion of the students of Midwest having a stereo system that cost more than or equal to $1,100 is less than 0.15.

Appropriate test: As the test is about testing the population proportion, the appropriate test is the Z-test for proportion and also the test is left tailed test with 0.05 level of significance.

Test statistic: The test statistic for the test is . Here is the sample proportion and the sample proportion is 57/300 = 0.19

Substitute the value in the test statistic.

Critical value: The test is left tailed test and so the left critical value at 5% level of significance is -1.6449.

Decision rule: The decision rule is to reject the null hypothesis if the observed test statistic is less than the left critical value -1.6449.

Conclusion: Here the observed test statistic is greater than the left critical value. So there is no evidence to reject the null hypothesis at 5% level of significance. It is reasonable to conclude that the population proportion of the students of Midwest having a stereo system that cost more than or equal to $1,100 is greater than or equal to 0.15. So the store can be opened in Midwest at 5% failure risk.

orchestra answered 2 years ago
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