91Ó°ÊÓ

A two-sample t-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. In this context, it helps us decide whether male and female basketball players have different mean approval ratings for unsportsmanlike play.
Here’s how it works:

• The test compares the sample means of two groups—in this case, male and female players.
• It uses sample data to infer information about population parameters (i.e., the actual mean approval ratings in all high school basketball players).
• The formula involves calculating a t-score, which tells us how far away the sample mean difference is from the hypothesized difference (0.5, in our case).

This test assumes that the two samples are independent, the data follows a normal distribution, and the variance in the two groups is approximately equal.
Applying the given data:

• Mean for males (\(ar{X}_m = 2.76\)) and females (\(ar{X}_f = 2.02\)).
• Standard deviations and sizes are used to calculate the t-score, which shows if the observed mean difference is statistically significant.

In hypothesis testing, it's essential to start by formulating two contrasting statements called the null and alternative hypotheses.
The **null hypothesis** (\(H0\)) assumes that there is no effect or difference. In this exercise, it proposes that the difference between the male and female approval rating means is less than or equal to 0.5. This is written as:\[\mu_m - \mu_f \le 0.5\]Where \(\mu_m\) and \(\mu_f\) stand for the mean approval scores for male and female players, respectively.
The **alternative hypothesis** (\(Ha\)) is what you want to prove. It opposes \(H0\), suggesting that the difference in mean ratings is greater than 0.5, expressed as:\[\mu_m - \mu_f > 0.5\]These hypotheses help determine the direction of the test. Here, we're doing a one-tailed test as we are checking if the male ratings exceed the female ratings by more than 0.5. Defining these hypotheses correctly ensures clarity in the testing process.

The significance level, often denoted as \(\alpha\), is a threshold that helps decide whether to reject the null hypothesis. It’s a probability measure that represents the risk of making a type I error—falsely rejecting a true null hypothesis.
In this exercise, \(\alpha = 0.05\), meaning there's a 5% risk of concluding that a difference exists when there's none. Choosing \(\alpha\) depends on how much risk you're willing to take. A common choice in many scientific disciplines is 0.05.
Once you perform the t-test, the resulting t-score is compared to a critical value from the t-distribution, which depends on \(\alpha\) and degrees of freedom (here, \(DF = 121\)). If your calculated t-score exceeds this critical value, you reject \(H0\).
Thus, the significance level is pivotal in hypothesis testing as it balances the risk of error and the need for evidence against the null hypothesis.