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When conducting hypothesis testing, one crucial concept is the test statistic. It's a standardized value that helps us determine whether to reject the null hypothesis. To calculate it, use the formula:

**\( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \)**.
Here’s how it works:

• \( \bar{x} \) represents the sample mean.
• \( \mu_0 \) is the population mean under the null hypothesis.
• \( s \) denotes the sample standard deviation.
• \( n \) is the sample size.

In our example, we calculate the test statistic using the given values:

**\( t = \frac{104.8 - 100}{9.2 / \sqrt{23}} \)**
After performing the calculations, we get a test statistic of approximately **2.503**. This number helps us determine if the sample mean is significantly different from the population mean.

Critical values are essential for drawing conclusions in hypothesis testing. These values define the boundaries of the rejection region in the tails of the distribution. For a given level of significance \( \alpha \) and degrees of freedom \( df \), you can find these values using statistical tables.

Let's break it down:

• We use a two-tailed test with \( \alpha = 0.01 \).
• The degrees of freedom are calculated as \( df = n - 1 \). With our sample size, \( df = 22 \).

Looking up the t-distribution table for \( \alpha / 2 = 0.005 \) and \( df = 22 \), we find the critical value to be **2.819**.
This means the two critical values for our test are **\( \pm 2.819 \)**. These values help demarcate the rejection regions of the distribution.

The t-distribution is a fundamental concept in hypothesis testing, especially when dealing with small sample sizes. It's similar to the normal distribution but has heavier tails, which accounts for the increased variability when estimating the population parameters. Here are some insights:

• It's especially handy when sample size is small (typically \( n < 30 \)).
• The shape of the t-distribution depends on degrees of freedom. As sample size increases, it approaches a normal distribution.

In our exercise, because the sample size is 23 and we're using a two-tailed test, we need to visualize the t-distribution with **22 degrees of freedom**. The critical regions fall in the tails, beyond our calculated critical values (\( t_{\alpha/2, df} = \pm 2.819 \)). If the test statistic falls within these tails, we reject the null hypothesis; otherwise, we do not. As we calculated earlier, our test statistic of **2.503** does not fall into these rejection regions, so we do not reject the null hypothesis. This approach ensures that we appropriately account for the sample size and variation in our hypothesis testing.