Question

Lengths of Prison Sentences A random sample of men and women in prison was asked to give the length of sentence each received for a certain type of crime. At \(\alpha=0.05,\) test the claim that there is no difference in the sentence received by each gender. The data (in months) are shown here. $$\begin{aligned}&\begin{array}{l|ccccccccc}\text { Males } & 8 & 12 & 6 & 14 & 22 & 27 & 32 & 24 & 26 \\\\\hline \text { Females } & 7 & 5 & 2 & 3 & 21 & 26 & 30 & 9 & 4\end{array}\\\&\begin{array}{l|ccccc}\text { Males } & 19 & 15 & 13 & & \\\\\hline \text { Females } & 17 & 23 & 12 & 11 & 16\end{array}\end{aligned}$$

Short answer

Males and females have significantly different sentence lengths.

Step-by-step solution

Gather Data and Hypotheses

List all data points for men and women: Males: 8, 12, 6, 14, 22, 27, 32, 24, 26, 19, 15, 13; Females: 7, 5, 2, 3, 21, 26, 30, 9, 4, 17, 23, 12, 11, 16. The null hypothesis is that there is no difference in the mean sentence lengths between genders: \( H_0: \mu_1 = \mu_2 \). The alternative hypothesis is that there is a difference: \( H_1: \mu_1 eq \mu_2 \).

Determine the Means and Standard Deviations

Calculate the means: Mean for males \( \bar{x}_1 = \frac{(8 + 12 + 6 + 14 + 22 + 27 + 32 + 24 + 26 + 19 + 15 + 13)}{12} = 17.5 \). Mean for females \( \bar{x}_2 = \frac{(7 + 5 + 2 + 3 + 21 + 26 + 30 + 9 + 4 + 17 + 23 + 12 + 11 + 16)}{14} = 13.071 \). Compute the standard deviations for both samples.

Calculate the Test Statistic

Use the two-sample t-test formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \], where \( s_1 \) and \( s_2 \) are the standard deviations and \( n_1, n_2 \) are the sample sizes. Calculate the variances: \( s_1^2 \) for males and \( s_2^2 \) for females, then compute the test statistic \( t \).

Determine Degrees of Freedom and Critical Value

The degrees of freedom for a two-sample t-test can be approximated using the formula: \[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]. Find the critical t-value from the t-distribution table at \( \alpha = 0.05 \) for the calculated degrees of freedom. This is a two-tailed test.

Compare Test Statistic to Critical Value

Check if the absolute value of the test statistic calculated in Step 3 exceeds the critical t-value obtained in Step 4. If \( |t| > t_{\text{critical}} \), reject the null hypothesis \( H_0 \). Otherwise, fail to reject \( H_0 \).

Conclusion

Based on the comparison in Step 5, decide the result. If the null hypothesis is rejected, it indicates that the difference in sentence lengths between genders is statistically significant.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis, often denoted as \( H_0 \), is a statement that posits no effect or no difference in the context of the study being conducted. It serves as the starting point for testing. For the exercise about prison sentences, the null hypothesis states that there is no difference in the average sentence lengths for males and females. Mathematically, this can be expressed as \( H_0: \mu_1 = \mu_2 \). This means that the mean length of sentences received by males (\( \mu_1 \)) is equal to that received by females (\( \mu_2 \)).

The purpose of the null hypothesis is to establish a baseline scenario that is tested statistically. It is assumed to be true until evidence suggests otherwise. In many tests, the goal is to either reject or fail to reject the null hypothesis based on data analysis. If the null hypothesis is rejected, it implies that the observed data show statistically significant differences or effects that indicate an alternative scenario.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_1 \), is the statement that contradicts the null hypothesis. It suggests that there is a significant effect or difference that needs attention. Unlike the null, the alternative hypothesis is what researchers aim to provide evidence for. In the context of the prison sentence lengths exercise, the alternative hypothesis claims there is a difference in the mean sentence lengths between genders: \( H_1: \mu_1 eq \mu_2 \).

The alternative hypothesis can be two-sided or one-sided. A two-sided hypothesis, like the one in this exercise, looks for any difference between the groups, whether males have longer or shorter sentences than females. Researchers conduct statistical tests to see if data provide enough evidence to support the alternative hypothesis over the null hypothesis. Proving \( H_1 \) typically requires demonstrating statistically significant results at a predetermined confidence level.

Significance Level

The significance level, denoted by \( \alpha \), is a critical threshold used in hypothesis testing. It represents the probability of falsely rejecting the null hypothesis, also known as the risk of a Type I error. In simpler terms, it is the cutoff point where differences observed in data are considered significant enough to disregard as mere chance.

For the prison sentence lengths, the significance level is set at \( \alpha = 0.05 \). This means that there is a 5% risk of rejecting the null hypothesis if it were actually true. A 5% significance level is a common choice; it reflects a balance between minimizing risk while being sensitive enough to detect true differences should they exist.

• Lower \( \alpha \) values (e.g., 0.01) indicate stricter criteria for significance, reducing Type I errors but potentially increasing Type II errors (failing to reject a false null).
• Higher \( \alpha \) values could lead to more frequent false positives but are less stringent.

Ultimately, choosing a significance level reflects the research goals and how critical it is to minimize errors.

Degrees of Freedom

Degrees of freedom (df) is a statistical concept representing the number of observations in a dataset that are free to vary. In the context of t-tests, degrees of freedom are crucial for determining the appropriate distribution to assess whether the test statistic is significant. It affects the shape and spread of the t-distribution used to obtain critical values.

For the comparison of prison sentences, the degrees of freedom is calculated using a formula involving variance and sample sizes for two independent groups:
\[df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}.\]

Here, \( s_1^2 \) and \( s_2^2 \) are sample variances and \( n_1, n_2 \) are sample sizes of males and females, respectively. The calculated degrees of freedom helps identify the precise critical t-value, aiding in deciding if the test statistic indicates significant differences. Accurately calculating and interpreting degrees of freedom ensures appropriate statistical conclusions drawn from the t-test results.