Question

Historically, 20% of graduates of the engineering school at a major university have been women. In a recent, randomly selected graduating class of 210 students, 58 were females. Does the sample data present convincing evidence that the proportion of female graduates from the engineering school has shifted (changed)? Use α = 0.05.

***I ALREADY HAVE A-E ANSWERED***
A.State the null and alternative hypotheses to be tested and indicate whether the test is left-tailed, right-tailed or two-tailed.

B. List the conditions that should be met in order to proceed with the hypothesis test and explain why (or show how) they are met.

C.Compute the test statistic and p-value for the hypothesis test and sketch the distribution of the test statistic, if the null hypothesis is true. Identify - label and shade - the region(s) represented by the p-value. Show your calculation(s).

D.Make a statistical decision about the null hypothesis (i.e. fail to reject H0 or reject H0), using the p-value approach. Justify your answer.

E.Write your conclusion in the context of the problem.

***I NEED F AND G ANSWERED***

F.Suppose we were to instead use a confidence interval to test if the proportion of female graduates from the engineering school differs from 20%.

-What would be the confidence level?

-Construct the confidence interval and explain how it supports your decision/conclusion made in (d) and (e). Show your calculation(s).

G.Determine the critical value(s) for this hypothesis test and explain how you would use it to come to the same decision/conclusion.

Answer 1

Solution:

Given: 20% of graduates of the engineering school at a major university have been women.

That is: p = 0.20

In a recent, randomly selected graduating class of 210 students, 58 were females.

Thus n = 210 and x = 58

Then sample proportion of women =

Level of significance = α = 0.05.

Part F) Suppose we were to instead use a confidence interval to test if the proportion of female graduates from the engineering school differs from 20%.

-What would be the confidence level?

As we have level of signficance = then

confidence level =

That is: 95% confidence level.

Construct the confidence interval and explain how it supports your decision/conclusion made in (d) and (e).

Formula:

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Thus

Since confidence interval is ( 0.2157 , 0.3367 ) is greater than the population proportion 0.20, we conclude that: the sample data present convincing evidence that the proportion of female graduates from the engineering school has shifted (changed).

Part G.Determine the critical value(s) for this hypothesis test and explain how you would use it to come to the same decision/conclusion.

level of signficance = ,

thus find Area =

From z table z critical value for area =0.9750 is 1.96

Since this is two tailed test , z critical values are: ( -1.96 , 1.96)

Decision rule: Reject H0 , if absolute z test statistic value > 1.96, otherwise we fail to reject H0.

From part c) test statistic value = z = 2.76

Since z test statistic value = 2.76 > z critical value = 1.96, we reject null hypothesis H0 and thus we conclude that: the sample data present convincing evidence that the proportion of female graduates from the engineering school has shifted (changed).