Question

Using Confidence Intervals

a. Assume that we want to use a 0.05 significance level to test the claim that p1 < p2. Which is better: A hypothesis test or a confidence interval?

b. In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; P-value method; critical value method?

c. If we want to use a 0.05 significance level to test the claim that p1 < p2, what confidence level should we use?

d. If we test the claim in part (c) using the sample data in Exercise 1, we get this confidence interval: -0.000508 < p1 - p2 < - 0.000309. What does this confidence interval suggest about the claim?

Short answer

a. A hypothesis test is better than a confidence interval.

b. The p-value method and critical value method are equivalent.

c. The 90% confidence interval to test the claim is made about the difference between two population proportion \({p_1}{\rm{ and }}{p_2}\).

d. The given confidence interval suggests that there is sufficient evidence to support the claim that \({p_1} < {p_2}\).

Step-by-step solution

Describe the methods of Inferences about two proportion.

a.The following methods can be used for comparing two population proportion:

1. Hypothesis test

2. Confidence Interval

The hypothesis test is used to test the claims about two population proportions. The confidence interval is used when estimating about the differences between two population proportions.

A hypothesis test is recommended for testing a claim at 0.05 level of significance.

Hence, in this case hypothesis test is better than confidence interval.

Check equivalency of three methods

b.In a hypothesis test, there are two methods to test the claims about population proportion,

1. P-value method

2. Critical value method

Both methods are used to test the claim made about two proportions. In p-value method, a probability value is compared to a significance level to make a decision while in critical value method, the test statistic is compared to critical value(s) to make a decision about hypotheses.

The method of confidence interval is used primarily to estimate the difference between two population proportions.

Thus, the P-value method and critical value method are equivalent.

State the confidence level

c. As per the table 8-1, for one tailed significance test at 0.05 level of significance, 90% confidence level is recommended.

Step 4:Make conclusion about the claim

d. Refer to exercise 1 for the claim stated as,

\(\begin{array}{l}{H_o}:{p_1} = {p_2}\\{H_a}:{p_1} < {p_2}\end{array}\)

Where \({p_1},{p_2}\) are population proportion of children who developed polio after vaccine and in control group respectively.

The given confidence interval is\( - 0.000508 < {p_1} - {p_2} < - 0.000309\).

This confidence does not contain zero. Thus, there is significant difference between two proportion\({{\rm{p}}_{\rm{1}}}{\rm{ and }}{{\rm{p}}_{\rm{2}}}\).

Therefore, the given confidence interval suggests that there is sufficient evidence to support the claim that \({p_1}{\rm{ }}\)is less that \({p_2}\).