(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. are my answers correct?

Question

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.

 1.      It is claimed that 70% of SHS students in a certain district have internet at home. A survey among 1500 SHS students in that district revealed that 1025 have internet at home. Use 0.05 level of significance

Are these correct?- https://imgur.com/XnLuoz4

https://imgur.com/GqTzy3b

 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.

Are these correct?- https://imgur.com/YoszwAG

https://imgur.com/LeStWFR

Answer 1

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1.      It is claimed that 70% of SHS students in a certain district have internet at home. A survey among 1500 SHS students in that district revealed that 1025 have internet at home. Use 0.05 level of significance

Ho: p = 0.7

Ha: p != 0.7

phat = 1025/1500 = 0.68

z = (0.68 - 0.7)/sqrt(0.7 * 0.3/1500) = -1.69

critical values for two-sided test and 0.05 level of significance = -1.96 and 1.96

reject Ho if |z| > 1.96. Here |z| = 1.69, so will not reject Ho.

there is not sufficient evidence to suggest the percentage of students having internet at home is different from 70% at the 0.05 level of significance

 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.

Are these correct?-

if no one prefers Brand A or Brand B we would suspect that the proportion that prefer either would be 0.5.

Ho: p = 0.5

Ho: p > 0.5

where p is proportion that prefer Brand A.

phat = 55/100 = 0.55

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

critical values for two-sided test and 0.05 level of significance = -1.96 and 1.96

reject Ho if |z| > 1.96. Here |z| = 1, so will not reject Ho.

there is not sufficient evidence to suggest that Brand A is preferred over Brand B at the 0.05 level of significance