Question

Two different simple random samples are drawn from two different populations. The first sample consists of 30 people with 14 having a common attribute. The second sample consists of 1900 people with 1370 of them having the same common attribute. Compare the results from a hypothesis test of p 1equalsp 2 ​(with a 0.01 significance​ level) and a 99​% confidence interval estimate of p 1minusp 2. What are the null and alternative hypotheses for the hypothesis​ test? A. Upper H 0​: p 1less than or equalsp 2 Upper H 1​: p 1not equalsp 2 B. Upper H 0​: p 1not equalsp 2 Upper H 1​: p 1equalsp 2 C. Upper H 0​: p 1equalsp 2 Upper H 1​: p 1less thanp 2 D. Upper H 0​: p 1equalsp 2 Upper H 1​: p 1not equalsp 2 E. Upper H 0​: p 1equalsp 2 Upper H 1​: p 1greater thanp 2 F. Upper H 0​: p 1greater than or equalsp 2 Upper H 1​: p 1not equalsp 2 Identify the test statistic. nothing ​(Round to two decimal places as​ needed.) Identify the critical​ value(s). nothing ​(Round to three decimal places as needed. Use a comma to separate answers as​ needed.) What is the conclusion based on the hypothesis​ test? The test statistic is ▼ not in in the critical​ region, so ▼ reject fail to reject the null hypothesis. There is ▼ sufficient insufficient evidence to conclude that p 1not equalsp 2. The 99​% confidence interval is nothingless thanleft parenthesis p 1 minus p 2 right parenthesisless than nothing. ​(Round to three decimal places as​ needed.) What is the conclusion based on the confidence​ interval? Since 0 is ▼ not included included in the​ interval, it indicates to ▼ reject fail to reject the null hypothesis. How do the results from the hypothesis test and the confidence interval​ compare? The results are ▼ the same different ​, since the hypothesis test suggests that p 1 ▼ greater than equals less than or equals greater than or equals less than not equals p 2​, and the confidence interval suggests that p 1 ▼ greater than or equals less than greater than less than or equals not equals equals p 2.

Answer 1

we have given that,

n1= number of peoples in 1st sample =30

x1=number of people in 1st sample having common attribute= 14

=1^st sample proportion = \frac{x1}{n1}=\frac{14}{30} =0.467

n2=number of peoples in 2nd sample=1900

X2=number of people in 2nd sample having common attribute=1370

= 2^nd sample proportion =\frac{x2}{n2}=\frac{1370}{1900} =0.721

Here we have given that,

Claim: To check whether the difference in two population proportions or not.

The Hypotheses is

v/s

Now, we can find the test statistics

= -2.78

We get

The Test statistic is -2.78

Now

P-value = 2*P( Z< -2.78)

              = 2 * ( 0.0027)) using standard normal z table

              =0.0054

we get P-value is 0.0054

Decision:

Here Pvalue < 0.01

Conclusion:

That is here there is sufficient evidence that the difference in two population proportions.

Now, we want to find the 99% confidence interval for the difference in two population proportion.

Formula is as follows,

Now we find the Z critical

Zcritical==2.58 ( using Excel=NORMSINV(PROB=0.01/2)

Now,

99 % confidence interval is

Interpretation:

This confidence interval shows that we are 99 % confident that the difference in the two population proportion is lies within that interval.