Question

A poll of two years ago, conducted by Louis Harris & Associates for Business Week magazine, examined the
perception of American adults of their government. When asked whether the U.S. government represents the
concerns of the American people or those of special interest groups, 63% of those polled felt that the U.S.
government primarily represents the concerns of special interest groups. Assume that this percentage was
true for the population of all American adults at the time the poll was conducted. A recent random sample
of 1200 American adults found that 61% of them feel that the U.S. government represents the concerns of
special interest groups. Let p be the percentage of all American adults who hold this view recently. Answer
the following questions.
(a) What is the point estimate for the p? What is the margin of error?
(b) Construct a 96% confidence interval for p. Is p included in this interval? What is the confidence based
on?
(c) Is the currant proportion of all U.S. adults who hold this opinion is less than 63%? Perform the hypothesis
testing at 1% significance level by answering the following questions.
i. Formulate the null and alternative hypothesis.
ii. State the test statistic and its approximate distribution.
iii. Determine the critical value for α = .01 and state the decision rule.
iv. Calculate the observed value of the test statistic from the sample.
v. State whether H0 is rejected and tell why.
vi. Calculate the P-value and do the hypothesis testing based on the P-value.
vii. Express the conclusion in the context of the problem, using common language

Answer 1

(a)

Point estimate of p = 0.61

Margin of error = = = 0.02892

(b)

YES, p is in the interval.

c) Hypothesis testing

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p=0.63

Ha: p<0.63

This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.

(2) Test Statistics

The z-statistic is computed as follows:

(3)Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a left-tailed test is zc​=−2.33.

The rejection region for this left-tailed test is R={z:z<−2.33}

(4)

test statistic = -1.435 as calculated above

(5) and (6) Decision about the null hypothesis

Since it is observed that z=−1.435≥zc​=−2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0756, and since p=0.0756≥0.01, it is concluded that the null hypothesis is not rejected.

(7) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pp is less than p0​, at the α=0.01 significance level. Hence we can conclude that the percentage of all Americans who hold the view that government primarily represents the concerns of special interest group is 63%.  

Graphically

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