Two different simple random samples are drawn from two different populations. Th

Question

Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute.

Conduct the hypothesis test that the two proportions are equal. Use a 0.05 level of significance. Next, construct the 95% confidence interval estimate of p1-p2. Summarize your conclusions for both analyses. If your conclusions differ in any way, explain why the results are different. Do you consider any difference between these results to be important? Justify your answer.

2.Goodness-of-fit Test: Making “cents” from checks or credit card purchases.

Consider how you would select a random sample of 100 purchases from your checkbook or 100 credit card purchases that you have made in the past year. (Do not mix checks and credit card purchases; pick one or the other for this study.) Describe how you know this is a random sample.

What population does your proposed sample describe? Would it be representative of most check or credit purchases in the U.S.? Why or why not?

Would you expect the purchases, classified by the “cents” as described by the categories below, to be evenly distributed? Explain why or why not.

Amount of Cents portion

0-0.24

0.25-0.49

0.50-0.74

0.75-0.99

Number of checks or credit card purchases

Collect your sample, as proposed, and summarize in the table above.

Use a .05 significance level to test the claim that the four categories are equally likely.

Did your results support your expectation? Provide an explanation.

Amount of Cents portion

0-0.24

0.25-0.49

0.50-0.74

0.75-0.99

Number of checks or credit card purchases

Explanation / Answer

H0 : p1 = p2

Ha : p1 p2

p^1 = 10/20 = 0.5

p^2 = 1404/2000 = 0.702

Here pooled estimate p = (10 + 1404)/ (20 + 2000) = 0.7

standard error of proportion = sqrt [p * (1 -p)* (1/n1 - 1/n2)] = sqrt [ 0.7 * 0.3 * (1/20 + 1/2000)] = 0.1030

Here test statistic

Z = (p^1 - p^2)/ se0 = (0.5 - 0.702)/ 0.1030 = -1.9615

So Z(critical) = - 1.96

so here as Z > Zcritical so we shall reject the null hypothesis and claim that both population proportion are different .

95 % confidence interval estimate =  (p^1 - p^2) +- Zcritical * sqrt [p1 * q1/n1 + p2q2/n2]

= (0.5 - 0.702) +- 1.96 * sqrt [0.5 * 0.5/20 + 0.702 * 0.298/2000]

= (-0.422, 0.018)

so here the interval consists the value of zero. SO we cannot reject the null hypothesis ere.

Here the results are different because of vast difference in two sample sizes.

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