91Ó°ÊÓ

In hypothesis testing, the test statistic helps us determine whether to reject the null hypothesis. For our exercise, where the goal is to test if the sample mean \(\bar{x}\) significantly exceeds a population mean \(\bar{\textbf{y}}\textbf{ }=40\), we use the formula for the t-test statistic for a sample mean:

\[ t = \frac{\bar{x} - \textbf{m}}{\frac{\textbf{s}}{\textbf{√s}}} \]
Given:
\[ \bar{x} = 42.3 \textbf{, }\textbf{u} = 40\textbf{, }\textbf{s} = 4.3 \textbf{, }\textbf{n} = 25 \]
Substituting these values gives us:
\[ t = \frac{42.3 - 40}{\frac{4.3}{\textbf{5}}} = \frac{2.3}{0.86} ≈ 2.674 \]
This calculated t-value of approximately 2.674 tells us how many standard errors the sample mean is from the hypothesized population mean. The larger the absolute value of the test statistic, the stronger the evidence against the null hypothesis.

The significance level, denoted by \( \textbf{α} \) (alpha), is the threshold we set for determining whether the test statistic is extreme enough to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true. Common significance levels include 0.05, 0.01, and 0.1.

In our exercise, the researcher decided on a significance level \( \textbf{α} = 0.1 \). This means that there is a 10% risk of concluding that the population mean is greater than 40 when it actually isn't. The choice of \( \textbf{α} \) affects the critical value against which the test statistic is compared. A lower \( \textbf{α} \) would require stronger evidence to reject the null hypothesis, making it less likely to make a Type I error (false positive).

The critical value marks the boundary beyond which we reject the null hypothesis. For a t-test, this value comes from the t-distribution table and depends on both the significance level \( \textbf{α} \) and the degrees of freedom \( df \).

For our scenario, with \( \textbf{α} = 0.1 \) and \( df = 24 \) (since \( df = \textbf{n} - 1 = \textbf{25} - 1 \)), the critical value is approximately 1.318 for a one-tailed test.

This means that if our test statistic is greater than 1.318, we can reject the null hypothesis. We set the critical region to the right of 1.318 on the t-distribution curve, indicating strong evidence against the null hypothesis. If our test statistic falls within this critical region, it suggests that the sample mean is significantly greater than the hypothesized population mean of 40.

Degrees of freedom (df) are crucial in determining the critical value from the t-distribution table. They relate to the sample size and provide an adjustment based on the information available from the data. Generally, for a single sample t-test, the degrees of freedom are calculated as:

\( df = n - 1 \)

Where \( n \) is the sample size.

In our exercise, the sample size is 25, so:

\[ df = 25 - 1 = 24 \]

These degrees of freedom help us find the appropriate critical value for our desired \( \textbf{α} \). Having 24 degrees of freedom means we look up the t-distribution table against this df to find the critical value for the given significance level, ensuring our hypothesis test is accurately calibrated. The higher the degrees of freedom, the closer the t-distribution mimics the normal distribution, reflecting the availability of more data.