Question

Students in Greek organizations at schools throughout the country sent volunteers to a yearly blood drive. The numbers from each randomly selected participating school are listed. Test the claim that there is no difference in the number of students participating from fraternities and sororities at \(\alpha=0.10\). $$\begin{array}{l|l}\text { Fraternities } & 4,5,10,7,7,15,12,11,13,15,12,12 \\\\\hline \text { Sororities } & 3,5,6,7,4,7,10,9,9,14\end{array}$$

Short answer

There is insufficient evidence to reject the null hypothesis at \( \alpha = 0.10 \); no significant difference in participation.

Step-by-step solution

Define the Hypotheses

The null hypothesis (H0) states that there is no difference in the number of students participating from fraternities and sororities: \( H_0: \mu_1 = \mu_2 \). The alternative hypothesis (H1) says there is a difference: \( H_1: \mu_1 eq \mu_2 \).

Calculate Sample Means and Standard Deviations

Calculate the means for each group: Fraternity mean \( \bar{x_1} \) and Sorority mean \( \bar{x_2} \), and their sample standard deviations \( s_1 \) and \( s_2 \).

Use the Two-Sample t-Test Formula

Use the formula for the two-sample t-test: \[ t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \] where \( n_1 \) and \( n_2 \) are the number of samples in each group.

Determine the Degrees of Freedom and Critical t-Value

Calculate the degrees of freedom using the formula: \[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1-1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2-1}} \]. Then, retrieve the critical t-value from t-distribution table at \( \alpha = 0.10 \) for a two-tailed test.

Compare and Conclude

Compare the calculated t-value with the critical t-value. If the calculated t-value is greater, reject the null hypothesis; otherwise, fail to reject it.

Key concepts

Two-Sample t-Test

In the world of statistics, a two-sample t-test is a method used to determine if there are significant differences between the means of two independent groups. This test is particularly useful when comparing data like the number of fraternity and sorority participants in the exercise example. The core idea is to see if there is evidence that the two groups have different average values or if any observed differences happened by chance.
The two-sample t-test answers the question: Are the observed differences in sample means reliable enough to reflect a real difference in the populations from which the samples were drawn? Here's how you tackle it:

• Gather the data from the two groups - in this case, the number of participants from fraternities and sororities.
• Calculate the sample means: \( \bar{x_1} \) for fraternities and \( \bar{x_2} \) for sororities.
• Use the standard deviations of both samples, \( s_1 \) for fraternities and \( s_2 \) for sororities, in the test formula.
• Apply the two-sample t-test formula to compute the t-score.

This t-score helps us understand how the samples relate to the hypothesized population means and if group differences are statistically significant.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It assumes that there is no significant difference between the specified populations. In the given exercise, the null hypothesis \( H_0 \) is expressed as \( \mu_1 = \mu_2 \), suggesting that fraternities and sororities have the same average number of participants attending the blood drive.
Formulating this hypothesis involves setting a baseline that any observed differences in sample data can be attributed to random variation, rather than a real effect. The strength of hypothesis testing lies in its ability to reject the null hypothesis, therefore suggesting that differences do exist.
Whenever conducting a hypothesis test:

• Step 1: Clearly state the null hypothesis \( H_0 \).
• Step 2: State the alternative hypothesis \( H_1 \), which indicates that a difference does exist.

These steps set the stage to utilize statistical calculations, compare results, and make informed conclusions.

Degrees of Freedom

Degrees of freedom are a critical part of determining the credibility of test statistics in hypothesis testing. It reflects the number of independent values or quantities which can vary in a data set. The formula for degrees of freedom when comparing two independent sample means is rather complex:
\[ df = \frac{ \left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]

• This formula helps account for the variability and size of the two groups being compared.
• Larger sample sizes typically increase the degrees of freedom, which can lead to more reliable results.

Knowing the degrees of freedom allows us to reference the correct t-distribution table to find the critical t-value. Hence, it plays a vital role when verifying if results are significant enough to reject the null hypothesis.

Critical t-Value

The critical t-value is part of the two-sample t-test and serves as a threshold to determine whether the observed test statistic (t-value) is extreme enough to reject the null hypothesis. This value depends on the degrees of freedom and the level of significance.
In the current exercise:

• A single critical t-value is extracted from the t-distribution table.
• The level of significance is given as \( \alpha = 0.10 \), indicating a 10% risk of concluding that a difference exists when there is none (Type I error).
• For a two-tailed test like ours, both upper and lower critical t-values are considered.

Once you have the critical t-value, compare it to the computed t-value from your test:

• If the absolute t-value calculated from your data exceeds the critical t-value, the null hypothesis is rejected.
• If not, there's insufficient evidence to claim a significant difference.

Understanding the relationship between the observed and critical t-values guides researchers to make sound decisions in hypothesis testing, highlighting real differences in populations.