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Chapter 11: Problem 22

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

Expert verified
Yes, based on these data and using a significance level of 5%, we have sufficient evidence to conclude that the mean cholesterol level is lower for persons who have attempted suicide compared to other patients.

Step by step solution

01

State the Hypotheses

First, let's state the null hypothesis and the alternative hypothesis. We are investigating whether the mean cholesterol level is lower for those who have attempted suicide compared to other patients. Therefore, our null hypothesis (H0) would be that the mean cholesterol level of the suicide attempt group (渭1) is equal to the mean cholesterol level of the other patients group (渭2). The alternative hypothesis (H1) would be that 渭1 is less than 渭2. In symbols, these would be:\n\nH0: 渭1 = 渭2\nH1: 渭1 < 渭2
02

Calculate the Test Statistic

Using the given sample means, standard deviations, and sample sizes, we can compute the Z score using the formula Z = (X虅1 - X虅2) / sqrt((蟽1^2/n1) + (蟽2^2/n2)), where X虅1 = 198, X虅2 = 217, 蟽1 = 20, 蟽2 = 24, n1 = n2 = 331. After substituting these values into the formula, we calculate the Z score to be approximately -15.87.
03

Find the Critical Value

Next, for a one-sided test at the 5% significance level (伪 = .05), we look up the critical value for z from a standard normal distribution table or use a calculator with z-distribution capabilities. The critical z value for 伪 = .05 (one-sided) is approximately -1.645.
04

Make a Decision

The rule of decision is: If the calculated absolute Z score is greater than the critical Z value, we reject the null hypothesis. Here, the absolute value of our Z score (-15.87) is indeed greater than our critical value (-1.645). Therefore, we would reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cholesterol Levels
Cholesterol levels in the blood are an important indicator of cardiovascular health. Cholesterol is a fatty substance found in the blood that is essential for building cells, but too much of it can lead to health problems. Generally, cholesterol comes in two forms:
  • LDL (Low-Density Lipoprotein): Often referred to as "bad" cholesterol because it can cause a buildup of fat in the arteries.
  • HDL (High-Density Lipoprotein): Known as "good" cholesterol as it helps remove LDL cholesterol from the arteries.
A balance of these is crucial for maintaining health, and typically, doctors recommend a blood cholesterol level of less than 200 mg/dL as ideal. The study mentioned focuses on comparing the cholesterol levels in two groups: patients who attempted suicide and those admitted for other reasons.
In the study, the group who attempted suicide had a mean cholesterol level of 198 mg/dL, while the other group had a mean of 217 mg/dL. These findings suggest a link between low cholesterol levels and certain mental health outcomes like depression and suicidal behavior, prompting further research into how cholesterol may affect mental health.
Standard Normal Distribution
The standard normal distribution is a vital concept in statistics that helps to understand how data points are spread in relation to the mean. It's a special type of normal distribution:
  • It is symmetric around the mean.
  • The mean of the distribution is 0.
  • The standard deviation is 1.
In hypothesis testing, we often convert sample means into Z scores by using the formula:\[Z = \frac{(X虅1 - X虅2)}{\sqrt{\frac{蟽1^2}{n1} + \frac{蟽2^2}{n2}}}\]where \(X虅1\) and \(X虅2\) are the sample means of the two groups, \(蟽1\) and \(蟽2\) are the standard deviations, and \(n1\) and \(n2\) are the sample sizes.
This standardization process helps us understand how extreme a particular test statistic is relative to the distribution of the null hypothesis. For instance, a calculated Z score of -15.87 indicates that the observed difference in cholesterol levels between the two groups is far from what we would expect if there were no true difference.
Statistical Significance
Determining statistical significance is key in hypothesis testing. It helps us to decide if the results observed in a study are likely due to some underlying cause or just random variation. A result is considered statistically significant if the probability of the observed outcome occurring by random chance is below a certain threshold, usually set at 5% (\(\alpha = 0.05\)).
To find out whether the results of this study on cholesterol levels are statistically significant, we compare the calculated Z score with the critical value derived from the standard normal distribution. With a Z score of -15.87 and a critical Z value of -1.645 for a one-sided test with \(\alpha = 0.05\), the absolute Z score far exceeds the critical value.
Thus, we reject the null hypothesis and conclude there is strong evidence to suggest that the mean cholesterol level is indeed lower for individuals who have attempted suicide compared to those who have not. This finding is statistically significant, underscoring an important connection that might influence future healthcare policies and management strategies for mental health.

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Most popular questions from this chapter

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