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.

2. A researcher claims that the distribution of the amounts that parents give for an allowance is different from the distribution shown in the pie chart. You randomly select 1103 parents and ask them how much they give for an allowance. The table shows the results. At α = 0.10, test the researcher’s claim. (Adapted from Echo Research)

Survey results
Response	Frequency, f
Less than $10	353
$10 to $20	167
More than $21	94
Don't give one/other	489

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 amounts differs from the expected distribution.

The hypothesis $H_0$ and $H_a$ is,

$H_0:$ The distribution of the allowance amounts is 29% less than $10, 16% $10 to $20, 9% more than $21 and 46% don’t give one/other.

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

Explanation of Solution

Given info:

The data shows the results of the distribution of the amounts that parent give for allowance.

Calculation:

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

The hypotheses are given below:

Null hypothesis:

$H_0:$ The distribution of the allowance amounts is 29% less than $10, 16% $10 to $20, 9% more than $21 and 46% don’t give one/other.

Alternative hypothesis:

$H_a:$ The distribution of amounts 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 6.251.

The rejection region is ${\chi }^2>6.251$.

Explanation of Solution

Given info:

The level of significance is 0.10.

Calculation:

Critical value:

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

Substitute k as 4 in degrees of freedom.

$4-1=3$

From the Table 6-Chi-Square Distribution, the critical value for 3 degrees of freedom for $\alpha =0.10$ level of significance is 6.251.

Rejection region:

The null hypothesis would be rejected if ${\chi }^2>6.251$.

Thus, the rejection region is ${\chi }^2>6.251$.

To determine

To obtain: The chi-square test statistic.

Answer to Problem 10.1.1RE

The chi-square test statistic is 4.886.

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 Response.
• 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 4.886.

To determine

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

Answer to Problem 10.1.1RE

The null hypothesis is fails to reject.

Explanation of Solution

Conclusion:

From the result of (c), the test-statistic value is 4.886.

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

That is, $4.886<6.251$.

Thus, it can be conclude that the null hypothesis fails to reject.

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 no evidence to support the claim that the distribution of amounts differs from the expected distribution.

Explanation of Solution

Interpretation:

From the results of part (d), it can be conclude that there is no evidence to support the claim that the distribution of amounts differs from the expected distribution.