HOMEWORK 17. H 0 : p = 0.50 H a : p b. Using the class data from the questionnaire, test your hypothesis.

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1 HOMEWORK Suose we select a radom samle of 1 studets ad fid that 43% said they believe i love at first sight. Which statemet is NOT ecessarily true? a. there were 43 studets i the samle who said they believe i love at first sight. b. based o the iformatio rovided by the samle, we caot determie exactly what roortio of the oulatio would say they believe i love at first sight. c. $.43 d I a February Gallu Poll, about 5% of the resodets said they believed i love at first sight. Let be the roortio of college studets who believe i love at first sight. a. What ull ad alterative hyotheses do you thik would be aroriate? H :.5 H a :.5 b. Usig the class data from the questioaire, test your hyothesis. 12m class: 25 out of 41 said they believe i love at first sight. That is the samle roortio is $ 25/ Coditios: 41( 5. ) 25. > 1 ad ) ) 25. > 1 Usig the formula, the test statistic is: z ) ) Usig the calculator: STAT TESTS 5: 1-ProZTest with.5, x 25, 41. The -value is.16. 2m class: 17 out of 42 said they believe i love at first sight. That is the samle roortio is $ 17/42.4. Coditios: 42( 5. ) 21 > 1 ad ) ) 21 > 1 Usig the formula, the test statistic is: z ) ) Usig the calculator: STAT TESTS 5: 1-ProZTest with.5, x 17, 42. The -value is.22

2 c. State your coclusio i cotext. For both classes: Sice the -value is greater tha 5%, or eve tha 1%, we do t have eough evidece to reject the ull hyothesis. That is, we do t have eough evidece to reject the claim that 5% of college studets believe i love at first sight. 3. Curretly, about 1% of marriages i the Uited States ed i divorce durig the first five years of marriage. A sociologist is studyig the effect of havig childre withi the first two years of marriage o the divorce rate. Usig hosital birth records, she selected a radom samle of 1 coules who had a child withi the first two years. Followig u o these coules, she fids that 17 of these coules had divorced withi the first five years. a. Set u aroriate ull ad alterative hyotheses to test whether havig childre withi the first two years of marriage affects the divorce rate. Assume havig childre could either icrease or decrease the divorce rate. H :.1 H a :.1 b. Calculate the value of the test statistic. 17 The samle statistic is $ Usig the formula, the test statistic is: z ) ) Usig the calculator: STAT TESTS 5: 1-ProZTest with.1, x 17, 1. c. Determie the -value of the test. -value:.196 d. Suose a test is erformed usig a sigificace level of α.5. Would we reject the ull hyothesis? What could we coclude, if aythig, about the effect of havig childre withi the first two years of marriage o the divorce rate? Sice -value <.5, we have eough evidece to reject the ull hyothesis. That is, we have eough evidece to reject the claim that 1% of marriages i the Uited States ed i divorce durig the first five years of marriage, ad we ca coclude that the this rate is ot 1%. 4. A uiversity admiistrator guesses that about 2% of studets at his college had set a text message durig class at least oce. The head of the Faculty Seate believes that the ercetage is higher tha that. I order to test the accuracy of the admiistrator s figure, a survey is coducted, ad 28% of the studets admitted to havig set a text message durig class at least oce. a. Write the ull ad alterative hyotheses i symbol otatio.

3 H :.2 Ha: >.2 b. Give the meaig of the arameter. is the roortio of ALL studets at that college who had set a text message durig class at least oce. c. What has to be true about the studet resoses i order for the test to have validity? Their resoses must be hoest, ad ubiased. d. What has to be true about the desig of the survey i order for the test to have validity? The samle must be a radom samle. e. What additioal iformatio would have to be rovided i order for you to coduct the hyothesis test? We eed to kow the samle size. f. The test was carried out, ad the resultig -value was.3. Fill i the blaks: If the true roortio of studets who have texted durig class is 2%, the the robability _ that a survey like this would result i 28% or more of the studets admittig to havig texted durig class is.3. g. Fill i the blaks: The -value give i (f) rovides moderately strog evidece that the roortio of studets who have texted durig class is higher tha 2%. h. At a sigificace level of 5%, would we reject the ull hyothesis? Justify your aswer. Yes. Sice the -value is less tha 5%, we have eough evidece to reject the ull hyothesis. That is, at the 5% level we ca reject the claim that the roortio of ALL studets at that college who had set a text message durig class at least oce is 2%. i. At a sigificace level of 1%, would we reject the ull hyothesis? Justify your aswer. No. Sice the -value is greater tha 1%, we do t have eough evidece to reject the ull hyothesis. That is, at the 1% level we caot reject the claim that the roortio of ALL studets at that college who had set a text message durig class at least oce is 2%. 5. A argumet broke out i a fraterity over who got to call the fli of a coi i a friedly bet. Oe of the frat members demaded to be allowed to make the call he wated to call heads because he believed that heads comes u more ofte tha tails. As a result of this argumet, aother member of the fraterity (who was afflicted with obsessive-comulsive disorder) tossed a coi 25, times to determie whether oe side teded to come u more ofte tha the other. He obtaied 12,671 heads. a. Determie a 95% cofidece iterval for the robability that the coi lads heads o a sigle toss. (This is the same as the log ru roortio of tosses that the coi would come u heads if it were tossed idefiitely.)

4 12671 The samle statistic is $ Coditios: 25(. 5684) > 1 ad ) ) > 1 The 95% cofidece iterval is: ) ± z * ± ) 25 ( 564., ) b. Use the cofidece iterval from (a) to test whether the coi comes u heads 5% of the time agaist the alterative that it does ot comes u heads 5% of the time. That is, H :.5 H a :.5 Sice.5 is ot i the CI, we would eed to reject the ull hyothesis. That is, we reject the claim that the roortio of heads is.5. c. What is the sigificace level of the test you carried out i art (b)? Sice we used a 95% CI, the sigificace level must be 5%. d. Perform the test i the usual way, by comutig the test statistic, determiig the -value, ad comarig it to the sigificace level. Use a sigificace level of 5%. Is your result cosistet with what you obtaied for art b? The test statistic is: z ) ) Usig the calculator STAT TESTS 5: 1-ProZTest with.5, x 12671, 25. The -value is.3 At the 5% level, we ca reject the ull hyothesis because the -value is less tha 5%. That is, we ca reject the claim that the roortio of heads is.5. We ca coclude that the coi is ot fair. This is the same coclusio as we had i art b. e. What would the -value have bee if we were testig our ull hyothesis agaist the alterative that the coi comes u heads more tha 5% of the time? Sice more tha is just oe tail, the -value would be half of the two-tailed test s -value, that is it would be.15. f. Based o your aswers to (b) ad/or (d), are the results statistically sigificat? Yes, based o the CI, ad the -value, the results are statistically sigificat.

5 g. Do the results have much ractical sigificace? Exlai. (Hit: The result of art (a) is articularly ertiet to this questio.) No, the results have ot much of a ractical sigificace. Sice the lausible values are betwee.564 ad.5134,.5 is PRACTICALLY right o the edge of the CI.