Music S.

asked • 05/21/22

 Mr. Santos, a garment store owner, claimed that the brand of sweatpants his customers prefer is Brand A compared to Brand B. He conducted a survey of 100 customers.

Direction: In the following problem, (a) state the null and alternative hypothesis, (b) select and compute the test statistic, (c) determine the critical value and the rejection region, and (d) draw a conclusion.


2.    Mr. Santos, a garment store owner, claimed that the brand of sweatpants his customers prefer is Brand A compared to Brand B. He conducted a survey of 100 customers. It was revealed that 55 prefer Brand A and the rest prefer Brand B. At 5% level of significance test the claim of Mr. Santos that the brand of sweatpants his customers prefer is Brand A compared to Brand B.

Jacob W.

tutor
First off Ive had to edit this explanation several times as the website does not seem to like symbols or formatting so this may be difficult to read. But I have typed out everything you need to answer the question. Any time you see 5 periods, that means there would have been a new paragraph there. Sorry this is so illegible but there is nothing I can do. Here it is: ....... Whats interesting about this question is that it can be adequately answered with two statistical techniques: Either the chi square goodness of fit test, or the z test of proportions. Which I would recommend you do depends on what section of your statistics class you are currently in (e.g. if youre currently reading the chapter on z tests, thats probably the test theyre expecting). For what its worth, the way this is written strikes me that they are expecting a z test so thats the one Ill cover. ........ For the z test we are going to test whether or not the true proportion is different than the hypothesized proportion. ........ The problem tells us the one of the hypotheses verbatim: Mr. Santos is stating the alternative hypothesis. What he is essentially saying is that the proportion of people who prefer Brand A is greater than 50%. I know this is an alternative hypothesis because an alternative hypothesis NEVER has an equal sign ( it can only contain Greater than, Less than, or not equal to) whereas a null hypothesis ALWAYS has some form of equal sign (it will contain Less than or equal to, greater than or equal to, or simply =). ...... Note that some classes teach that the null hypothesis always uses = regardless of direction. Your mileage may vary, but overall the reasoning is the same; Mr. Santos has to be stating the alternative hypothesis because he is making the prediction that A greater than 50% and the symbol greater than is ONLY ever used in alternative hypotheses. Also note that this website will not allow me to type the greater than and less than symbols, quote symbols, or apostrophes without glitching so you will have to make do with me typing them out. ...... Now that we know the alternative hypothesis, we can infer the null hypothesis. It is the opposite of Mr. Santos statement. The proportion of customers that prefer A is equal to or less than 50% (again some classes might claim it is simply A = 50% but they mean the same thing). ...... So to summarize so far H0: proportion who prefer A less than or equal to 50% HA: proportion who prefer A greater than 50% ...... Next we need to determine our decision rule. Alpha = 0.05. This is a one tailed test (if it was two tailed the alternative would be not=) so we DO NOT need to cut alpha in half at this step. We leave alpha as is. So lets look up what the equivalent z value for 0.05 is on the z table. The value on the inside of the z table that is closest to 0.05 is the z value of 1.64 or 1.65. This is a common one so I happen to know that we need to split the difference. Its 1.645. ....... Thats our critical value: Z-critical is 1.645. Because this is an upper tailed test we know that this value is going to be positive. If it were a lower tailed test it wouldve been negative (in that case the alternative hypothesis wouldve contained less than; rather than greater than). ...... So our decision rule is that the z test statistic must be higher than 1.645 to reject the null hypothesis. But what is the test statistic? ....... To get the test statistic we need to do some math. ...... Its difficult for me to write the formula here but here is a website I found that has it: https://www.leansigmacorporation.com/one-sample-proportion-test-minitab/ ...... P-hat is our observed proportion that preferred A: 0.55. ...... P-subzero is the proportion that the null hypothesis assumes, i.e. that A and B are equally preferred: A = 0.5. So P-subzero is 0.5. ....... n is the number of people in the sample: 100 ...... P-hat = 0.55 ...... P-subzero = 0.5 ...... n = 100 ...... so our numerator in the formula will be 0.55-0.5 = 0.05. ...... So now the denominator 0.5*(1-0.5) = 0.25 ....... 0.25/100 ....... 0.0025 ...... Square root of 0.0025 = 0.05 ...... So the denominator is 0.05. ...... Now we divide the numerator and the denominator 0.05/0.05 = 1 ....... Weird coincidence that the numerator and denominator worked out to be the same number but our answer is 1. ....... Z-test statistic = 1 ...... Z-critical value = 1.645 ....... This is not a significant enough difference. 1 is not greater than 1.645 so we fail to reject the null hypothesis. In plain english, this means there is not enough evidence to support Mr. Santos claim that customers prefer Brand A. ....... I did this rather quickly but Im fairly certain I did everything correctly. Let me know if this is correct or if any of this explanation was helpful! Good luck!
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05/21/22

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Sadia S. answered • 05/21/22

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Mr. Santos, a garment store owner, claimed that the brand of sweatpants his customers prefer is Brand A compared to Brand B. He conducted a survey of 100 customers. It was revealed that 55 prefer Brand A and the rest prefer Brand B. At 5% level of significance test the claim of Mr. Santos that the brand of sweatpants his customers prefer is Brand A compared to Brand B.




if neither is preferred then proportion preferring brand A is 0.5.




Ho: p = 0.5


Ha: p > 0.5




phat = 55/100 = 0.55




z = (0.55 - 0.5)/sqrt(0.5*0.5/100) = 1




rejection region for alpha = 0.05 and right sided test is z > 1.645




because z not in rejection region do not reject Ho

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