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Hypothesis Testing for Proportions Practice Questions

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Hypothesis Testing for Proportions Practice Questions

Practice

A woman woke up this morning feeling sick to her stomach. She is not sure whether or not she needs to see the doctor. She has no insurance, so she wants to be sure that she really needs medical attention before going to the doctor. Based on this, answer the following questions.

1. State the null and alternative hypotheses.
2. How could the woman make a type I error and what would the consequences be?
3. How could the woman make a type II error and what would the consequences be?
4. A large city is contemplating a ban on smoking in public spaces. The city council wants to institute such a ban only if more than 70% of the adults living in the city support it. To find out if such support exists, the city manager randomly selects 150 of the adult residents in the city and asks them whether or not they support the ban. Of the 150 citizens, 108 support the ban and 42 do not. Is there sufficient statistical evidence to conclude that the strong support for the ban is present among the city's residents? Be sure to follow the five steps of hypothesis testing.

Solutions

1. H0 = do not go to the doctor; Ha = go to the doctor.
2. A type I error would occur if the woman did go to the doctor when it really wasn't necessary, thus needlessly spending money.
3. A type II error would occur if the woman did not go to the doctor when she was really in need of medical attention, possibly causing her illness to get worse.
4. (i) The set of hypotheses to be tested are H0 : p = 0.70 versus Ha: p > 0.70.

(ii) The mayor took a random sample of adult residents of the city, so the first condition for inference is satisfied. To check the second condition, we have np = 150 () = 108 > 10 and n(1 – p) = 150(1 – ) = 42 > 10. Thus, the second condition for inference is also satisfied. The test statistic = 0.53.

(iii) The p-value is 0.2981.

(iv) We do not reject the null hypothesis. (v) There is not sufficient evidence to conclude that more than 70% of the city's adult residents favor a ban on smoking in public places.

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