Question

According to the Associated Press (San Luis Obispo Telegram-Tribune, June 23,1995 ), a study by Italian researchers indicated that low cholesterol and depression were linked. The researchers found that among 331 randomly selected patients hospitalized because they had attempted suicide, the mean cholesterol level was 198 . The mean cholesterol level of 331 randomly selected patients admitted to the hospital for other reasons was 217 . The sample standard deviations were not reported, but suppose that they were 20 for the group who had attempted suicide and 24 for the other group. Do these data provide sufficient evidence to conclude that the mean cholesterol level is lower for those who have attempted suicide? Test the relevant hypotheses using \(\alpha=.05\).

Short answer

The exact decision about whether the mean cholesterol level is lower for those who attempted suicide cannot be made without conducting the calculations in Steps 2 and 4. The steps provide a process to make a conclusion based on the difference in mean cholesterol levels for the two groups.

Step-by-step solution

State the Null and Alternative Hypotheses

The null hypothesis \(H_0\) would state that there is no difference in the cholesterol levels of the two groups. That is, the mean cholesterol level of patients who attempted suicide equals the mean cholesterol level of patients hospitalized for other reasons. So, \(H_0: \mu_{1} = \mu_{2}\). The alternative hypothesis \(H_a\) would state that the mean cholesterol levels are lower in patients who attempted suicide, so \(H_a: \mu_{1} < \mu_{2}\). Here, \(\mu_1\) and \(\mu_2\) represent the population mean cholesterol levels for the suicide attempt and other reasons groups respectively.

Calculate the Test Statistic

The test statistic for the difference between two means when the standard deviations are known is given by \(z = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{{s_1}^2}{n_1} + \frac{{s_2}^2}{n_2}}}\), where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes. Substituting the given data, we find the test statistic \(z\).

Determine the Rejection Region

The rejection region is determined based on the significance level, \(\alpha = .05\), and it is the set of values for which we reject the null hypothesis. Since the alternative hypothesis is \(H_a: \mu_{1} < \mu_{2}\), this is a one-tailed (left-tailed) test. The critical value \(z_{critical}\) can be found from the standard statistical table or using statistical computing software. If the calculated test statistic \(z\) falls in the rejection region (i.e., \(z < z_{critical}\)), then reject the null hypothesis.

Make the Decision

If the test statistic is in the rejection region, reject \(H_0\) and conclude there is sufficient statistical evidence that the mean cholesterol level is lower for those who attempted suicide. Else, do not reject \(H_0\) and conclude there is not enough evidence to make that conclusion.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is the very foundation of hypothesis testing in statistics. The null hypothesis (\( H_0 \)) represents the default claim, often suggesting that there is no effect or no difference between groups. In our case, the null hypothesis posits that the mean cholesterol level of patients who attempted suicide is the same as those who were hospitalized for other reasons, noted as \( H_0: \mu_{1} = \mu_{2} \). Conversely, the alternative hypothesis (\( H_a \) or \( H_1 \) ) represents what the researcher seeks to prove; it suggests there is a true effect or difference. For the cholesterol study, the alternative hypothesis claims that the mean cholesterol level is lower for those who attempted suicide \( (H_a: \mu_{1} < \mu_{2}) \). These hypotheses must be mutually exclusive and exhaustive.

Mean Comparison

Comparing means is a statistical technique used to evaluate the difference between two group means. It is a critical component when drawing conclusions from data such as in the study concerning cholesterol levels and patients' circumstances. In our case, we are comparing the mean cholesterol levels between two groups of patients—those who attempted suicide versus those hospitalized for other reasons. To see if the observed difference in means is statistically significant, we utilize tests like the z-test when standard deviations are known and sample sizes are large, or the t-test when standard deviations are unknown and/or sample sizes are smaller. This comparison must be handled with precision, as it informs the decision-making at the end of the hypothesis testing process.

Test Statistic Calculation

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It's meant to assess the strength of the evidence against the null hypothesis. To compute the test statistic in our example, where standard deviations are known, we use the z-test formula: \( z = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{{s_1}^2}{n_1} + \frac{{s_2}^2}{n_2}}} \). Once we plug in the sample means (\( \bar{x}_1 \) and \( \bar{x}_2 \) ), the sample standard deviations (\( s_1 \) and \( s_2 \) ), and the sample sizes (\( n_1 \) and \( n_2 \) ), we determine the z-value, which tells us how many standard deviations the sample mean difference is away from the null hypothesis's predicted mean difference.

Significance Level

The significance level, denoted as \( \alpha \), is a threshold used to determine when to reject the null hypothesis. In simpler terms, it represents the probability of making the mistake of rejecting the null hypothesis when it is actually true, known as a Type I error. A common choice for \( \alpha \) is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is no actual difference. In our exercise, a significance level of \( \alpha = .05 \) is chosen. This means that if the probability of observing the test statistic is less than 5% assuming that the null hypothesis is true, then we consider the result statistically significant, and the null hypothesis can be rejected.

One-tailed Test

A one-tailed test, or directional test, predicts specifically whether a parameter will be greater or less than the value stated in the null hypothesis. Such tests are used when the researcher has a hypothesis about the direction of the relationship or difference. For instance, as in our cholesterol levels study, if the alternative hypothesis suggests that one mean is less than the other (\( H_a: \mu_{1} < \mu_{2} \)), a one-tailed test is appropriate. The rejection region for a one-tailed test is only on one side of the distribution, which corresponds to the direction of the alternative hypothesis. In our exercise, it's a left-tailed test because the research predicts a 'lower than' relationship, thus, we are only concerned with values in the lower end of the z-distribution.