Chapter 10, Problem 10.1.1RE

In Exercises 1−4. (a) identify the claim and state H₀ and H_a, (b) find the critical value and identify the rejection region, (c) find the chi-square test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

1. A researcher claims that the distribution of the lengths of visits at physician offices is different from the distribution shown in the pie chart. You randomly select 400 people and ask them how long their office visits with a physician were. The table shows the results. At a = 0.01, test the researcher’s claim, (Adapted from Medscape)

Survey results
Minutes	Frequence, f
less than 9	20
10−12	80
13−16	113
17−20	91
21−24	40
25 or more	56

To determine

To identify: The claim.

To state: The hypothesis $H_0$ and $H_a$.

Answer to Problem 10.1.1RE

The claim is that, the distribution of the lengths differs from the expected distribution.

The hypothesis $H_0$ and $H_a$ is,

$H_0:$ The distribution of the lengths of visits at physician offices is not differs from the expected distribution.

$H_a:$ The distribution of the lengths differs from the expected distribution.

Explanation of Solution

Given info:

The data shows the results of the distribution of the lengths of the visits at physician offices.

Calculation:

Here, the distribution of the lengths differs from the expected distribution is tested. Hence, the claim is that the distribution of the lengths differs from the expected distribution.

The hypotheses are given below:

Null hypothesis:

$H_0:$ The distribution of the lengths of visits at physician offices is not differs from the expected distribution.

Alternative hypothesis:

$H_a:$ The distribution of the lengths differs from the expected distribution.

To determine

To obtain: The critical value.

To identify: The rejection region.

Answer to Problem 10.1.1RE

The critical value is 15.086.

The rejection region is ${\chi }^2>15.086$.

Explanation of Solution

Given info:

The level of significance is 0.01.

Calculation:

Critical value:

The critical value is calculated by using the $\left(k-1\right)$ degrees of freedom.

Substitute k as 6 in degrees of freedom.

$6-1=5$

From the Table 6-Chi-Square Distribution, the critical value for 5 degrees of freedom for $\alpha =0.01$ level of significance is 15.086.

Rejection region:

The null hypothesis would be rejected if ${\chi }^2>15.086$.

Thus, the rejection region is ${\chi }^2>15.086$.

To determine

To obtain: The chi-square test statistic.

Answer to Problem 10.1.1RE

The chi-square test statistic is 18.770.

Explanation of Solution

Calculation:

Step by step procedure to obtain chi-square test statistic using the MINITAB software:

• Choose Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable).
• In Observed counts, enter the column of Frequency.
• In Category names, enter the column of Minutes.
• Under Test, select the column of Proportions in Proportions specified by historical counts.
• Click OK.

Output using the MINITAB software is given below:

Thus, the chi-square test statistic value is approximately 18.770.

To determine

To check: Whether the null hypothesis is rejected or fails to reject.

Answer to Problem 10.1.1RE

The null hypothesis is rejected.

Explanation of Solution

Conclusion:

From the result of (c), the test-statistic value is 18.770.

Here, the chi-square test statistic value is greater than the critical value.

That is, $18.770>15.086$.

Thus, it can be conclude that the null hypothesis is rejected.

To determine

To interpret: The decision in the context of the original claim.

Answer to Problem 10.1.1RE

The conclusion is that, there is evidence to support the claim that the distribution of the lengths differs from the expected distribution.

Explanation of Solution

Interpretation:

From the results of part (d), it can be conclude that there is evidence to support the claim that the distribution of the lengths differs from the expected distribution.