Question

Doing Time. The Federal Bureau of Prisons publishes data in Statistical Report on the times served by prisoners released from federal institutions for the first time. Independent simple random samples of released prisoners for five different offence categories yielded the data on time served, in months, shown on the WeissStats site.

a. Obtain individual normal probability plots and the standard deviation of the samples.

b. Perform a residual analysis.

c. Use your results from parts (a) and (b) to decide whether conducting a one-way ANOVA test on the data is reasonable. If so. also do parts (d) and (e).

d. Use a one-way ANOVA test to decide, at the $5\%$significance level, Whether the data provide sufficient evidence to conclude that a difference exists among the means of the populations fewer than the samples were taken.

e. Interpret your results from part (d).

Short answer

(a)

(b)

(c) It appears acceptable to assume that the assumption of normal populations is met based on the normal probability plot of residuals.

(d)

(e) At the $5\%$significance level, the data provide sufficient evidence to conclude that a difference exists in the mean served times of the five offence categories.

Step-by-step solution

Part (a) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (a) Step 2: Explanation

The individual normal probability plots of the samples are given in the following MINITAB output:

The mean and standard deviation of the samples are given in the following MINITAB output:

Part (b) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (b) Step 2: Explanation

The residual analysis' normal probability is plotted, and the residual value increases as the percentage increases.

Part (c) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (c) Step 2: Explanation

It appears plausible to assume that the assumption of normal populations is met based on the normal probability plots.

We can see that the highest sample standard deviation $(4.700)$ is less than twice the lowest sample standard deviation from the standard deviations of the four samples (3.601). As a result, the criteria of equal standard deviations in the populations is met.

It appears acceptable to assume that the assumption of normal populations is met based on the normal probability plot of residuals.

Part (d) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (d) Step 2: Explanation

The null and alternative hypotheses are

$H_0:{\mu }_1={\mu }_2={\mu }_3={\mu }_4={\mu }_5$(Mean served times are equal)

$H_a:$Not all the means are equal

We are to perform the test at the $5\%$significance level;

so $\alpha =0.05$

The MINITAB output is as follows:

Part (e) Step 1: Concept introduction

ANOVA, or non - parametric tests, is a potent quantitative approach for revealing the disparity amongst two or more phenomena or aspects using variable importance. It also illustrates how to determine statistical significance among several demographics.

Part (e) Step 2: Explanation

Because the P-value $(0.000)$is smaller than the significance level we reject $H_0$.

Hence at the significance level the data provide sufficient evidence to conclude that a difference exists in mean served times of the five offense categories.