Question

Using Confidence Intervals to Test Hypotheseswhen analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

a.Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

b.Use the P-value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

c.Use the sample data to construct a 95% confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1?

d.Compare the results from the critical value method, the P-value method, and the confidence interval method. Do they all lead to the same conclusion?

Short answer

a. Using the critical value method, it is concluded that there is enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

b. Using the p-value method, it is concluded that there is enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

c. Using the confidence interval method, it is concluded that there is not enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

d. The p-value method and the critical method lead to the same conclusion of rejecting the claim that the population proportion of zeros is equal to 0.1. The confidence interval method leads to the conclusion of failing to reject the claim that the population proportion of zeros is equal to 0.1. All the methods do not lead to the same conclusion.

Step-by-step solution

Given information

In a sample of randomly selected 1000 digits, the number of digits which is zero is equal to 119. It is claimed that the proportion of zeros should be 0.1.

State the hypotheses

The hypotheses are as follows:

Null Hypothesis: The proportion of zeros is equal to 0.1.

\({H_0}:p = 0.1\)

Alternative Hypothesis: The proportion of zeros is not equal to 0.1.

\({H_1}:p \ne 0.1\)

Sample proportion and population proportions

The sample proportion of zeros is computed below:

\(\begin{array}{c}\hat p = \frac{{119}}{{1000}}\\ = 0.119\end{array}\)

The population proportion of zeros is given below:

\(p = 0.1\)

The population proportion of non-zero numbers is given below:

\(\begin{array}{c}q = 1 - p\\ = 1 - 0.1\\ = 0.9\end{array}\)

Compute the test statistic

The test statistic is computed as follows:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.119 - 0.1}}{{\sqrt {\frac{{0.1 \times 0.9}}{{1000}}} }}\\ = 2.003\end{array}\)

The test statistic value is 2.003.

Critical value method

a.

Referring to the standard normal distribution table, the critical values of z at\(\alpha = 0.05\)for a two-tailed test are -1.96 and 1.96.

The test statistic value does not lie within the two critical values. Thus, the null hypothesis is rejected.

There is enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

P-value method

b.

The two-tailed p-value of z equal to 2.003 is copmputed as follows:

\(\begin{array}{c}2P\left( {z > 2.003} \right) = 2\left( {0.0226} \right)\\ = 0.0452\end{array}\)

Therefore, the p-value of z equal to 2.003 is equal to 0.0452.

Since the p-value is less than 0.05, the null hypothesis is rejected.

There is enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

Confidence interval method

c.

The expression for computing the confidence interval is as follows:

\(\hat p - E < p < \hat p + E\)

The formula to calculate the value of the margin of error (E) is written below:

\(E = {z_{\frac{\alpha }{2}}} \times \sqrt {\frac{{\hat p\hat q}}{n}} \)

The confidence level is 95%. Thus, the corresponding level of significance is equal to 0.05.

The value of\({z_{\frac{\alpha }{2}}}\)when\(\alpha = 0.05\)is equal to 1.96.

The margin of error is computed as shown below:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}} \times \sqrt {\frac{{\hat p\hat q}}{n}} \\ = 1.96 \times \sqrt {\frac{{0.119 \times 0.881}}{{1000}}} \\ = 0.0201\end{array}\)

The confidence interval is computed below:

\(\begin{array}{c}\hat p - E < p < \hat p + E\\0.119 - 0.0201 < p < 0.119 + 0.0201\\0.0989 < p < 0.1391\end{array}\)

The 95% confidence interval estimate of the population proportion of zeros is equal to (0.0989, 0.1391).

Since the interval contains the value of 0.1, it can be said that the population proportion of zeros can be equal to 0.1.

Thus, there is not enough evidence to reject the claim that the population proportion of zeros is equal to 0.1.

Comparing the results obtained from the 3 methods

d.

The results from the p-value method and the critical method are the same and lead to the conclusion of rejecting the claim that the population proportion of zeros is equal to 0.1.

The result from the confidence method leads to the conclusion of failing to reject the claim that the population proportion of zeros is equal to 0.1.

Thus, all the methods do not lead to the same conclusion.