Question

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether people with a specific genetic marker are more likely to have suffered from clinical depression than people without the genetic marker, using the information that \(38 \%\) of the 42 people in a sample with the genetic marker have had clinical depression while \(12 \%\) of the 758 people in the sample without the genetic marker have had clinical depression.

Short answer

The relevant proportions are: for people with the genetic marker \(p_1 = 0.38\); for people without the genetic marker \(p_2 = 0.12\). The pooled proportion is \(p = 0.1237\). Using the normal distribution for hypothesis testing, the calculated Z value is approximately 2.348, which is larger than the critical value of 1.645. Therefore, the null hypothesis is rejected, providing evidence that people with the genetic marker are more likely to have suffered from clinical depression than those without the genetic marker.

Step-by-step solution

Determine Relevant Sample Proportions

First, find the relevant proportions. The proportion of people with the genetic marker who have had depression is \(0.38\), while the proportion of people without the marker who've had depression is \(0.12\). The number of people with the genetic marker (denoted as \(n_1\)) is 42 and the number of people without the genetic marker (denoted as \(n_2\)) is 758.

Compute Pooled Proportions

Next, calculate the pooled proportion. This means calculating a combined proportion for everyone in the study, regardless of whether or not they have the genetic marker. This is calculated as \((X_1+X_2) / (n_1+n_2)\), where \(X_1\) and \(X_2\) are the number of successes (here defined as incidences of depression) in each group, respectively. Here, \(X_1 = 0.38 \times 42 = 15.96\) and \(X_2 = 0.12 \times 758 = 90.96\). The pooled proportion (\(p\)) is thus \((15.96+90.96) / (42+758) = 0.1237\).

Set Up Hypothesis

The null hypothesis (\(H_0\)) is there is no difference in incidence of depression among people with and without the genetic marker (\(p_1 - p_2 = 0\)). The alternative hypothesis (\(H_a\)) is that those with the genetic marker are more likely to have depression (\(p_1 - p_2 > 0\)).

Hypothesis Test Using Normal Distribution

Perform a hypothesis test using normal distribution. The test statistic can be calculated using the formula for the difference of two proportions: \(Z = (p_1 - p_2 - 0) / \sqrt{ p ( 1-p ) [(1/n_1) + (1/n_2)] }\). Here, \(p_1 = 0.38\), \(p_2 = 0.12\), \(n_1 = 42\), \(n_2 = 758\), and \(p = 0.1237\). Upon calculation, \(Z\approx 2.348\). If this value is greater than the critical value for a given significance level (say 5%, for which \(Z_{\text{critical}} = 1.645\)), we would reject the null hypothesis.

Conclusion

Since \(Z > Z_{\text{critical}}\) (2.348 vs 1.645), we reject the null hypothesis. Therefore, based on the sample data and the hypothesis test, individuals with the specific genetic marker are significantly more likely to have suffered from clinical depression than those without the genetic marker.

Key concepts

Sample Proportions

Understanding sample proportions is crucial when working with statistics, especially in hypothesis testing. A sample proportion represents the ratio of individuals in a sample meeting a certain criterion to the total number of individuals in that sample. It is the estimate of a population proportion derived from a sample. For example, if we're investigating how many people suffer from a certain condition, like clinical depression, within a specific group, the sample proportion gives us an insight into the larger population's behavior.

When conducting a study on clinical depression and a genetic marker, you might find that a sample proportion of individuals with the genetic marker is 0.38 (38%), which means that of those sampled with the marker, 38% have had depression. Contrast this with a sample proportion of 0.12 (12%) for individuals without the genetic marker. This stark difference can be an indicator for further investigation through hypothesis testing.

Pooled Proportion

The concept of a pooled proportion comes into play when conducting hypothesis tests involving two sample proportions. It is essentially a weighted average of the sample proportions, where the weights are the sample sizes. This combined proportion assumes that there's no difference between the groups with regard to the characteristic being tested, which is a necessary assumption in some hypothesis tests.

Using our depression and genetic marker example, we calculate the pooled proportion by summing the number of individuals with and without the marker who have had depression, then dividing by the total number of individuals in both samples combined. The formula, \( (X_1 + X_2) / (n_1 + n_2) \), allows us to obtain a single rate that represents the overall incidence of depression in the entire sample group, without considering the presence of the genetic marker. It's a crucial step in hypothesis testing as it leads to the calculation of the expected frequencies and subsequently, the test statistic itself.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is a fundamental concept in statistics and forms the basis for various hypothesis tests, including the one used when testing for differences in sample proportions.

In hypothesis testing, the normal distribution allows us to determine how far away our test statistic is from what we would expect under the null hypothesis. This is done by converting the test statistic into a z-score, which represents how many standard deviations away from the mean a certain observation is. If our calculated z-score falls into the tail of the distribution beyond a certain critical value (determined by our chosen significance level), we can conclude there's a statistically significant difference between groups—in our case, between the depression rates of those with and without the genetic marker.

Genetic Marker and Depression

The relationship between genetic markers and depression is the basis for many important studies. A genetic marker is a gene or DNA sequence with a known location on a chromosome that can be used to identify individuals or species. It can serve as a signpost for a trait of interest, such as susceptibility to a disease like depression.

Hypothesis testing can be employed to determine if there is a significant relationship between a given genetic marker and the incidence of depression. By comparing the depression rates of individuals with and without the genetic marker, researchers can assess whether the presence of the marker correlates with a higher risk of the condition. In our exercise, the hypothesis test suggested a significant difference, indicating that the genetic marker may be associated with an increased likelihood of clinical depression. Such findings are essential for understanding the biological underpinnings of mental health conditions and can help inform targeted interventions and treatments.