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In hypothesis testing, we need to clearly define two hypotheses: the null hypothesis and the alternate hypothesis. These form the foundation of our test.

The **null hypothesis** (\( H_0 \)) proposes that there is no effect or difference, and in this exercise, it asserts that the population means from two different samples are equal. It is expressed as \( \mu_1 = \mu_2 \), suggesting that any observed difference is due to random chance.

Conversely, the **alternate hypothesis** (\( H_1 \)) suggests that there is indeed an effect or a difference between the two population means. It counterclaims the null hypothesis by stating \( \mu_1 eq \mu_2 \), indicating that the means are not equal.

• **Null Hypothesis (\( H_0 \)):** No difference.\( \mu_1 = \mu_2 \).
• **Alternate Hypothesis (\( H_1 \)):** There is a difference.\( \mu_1 eq \mu_2 \).

Understanding these hypotheses is crucial as they guide the selection of the statistical test and influence the interpretation of the results.

Calculating the test statistic is a key step in hypothesis testing, acting like the evidence you provide to support or refute your hypotheses. The test statistic tells you how many standard deviations your sample mean is away from the null hypothesis mean.

**Step-by-Step Test Statistic Calculation:**

• Calculate the **standard error** of the difference between the means.\[ SE = \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} } \] With given values: \( S_1 = 4 \), \( n_1 = 10 \), \( S_2 = 5 \), \( n_2 = 8 \),\[ SE = \sqrt{ \frac{4^2}{10} + \frac{5^2}{8} } = \sqrt{ 4.725 } \approx 2.173 \]
• Determine the **test statistic**:\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{23 - 26}{2.173} \approx -1.38 \]

This calculation combines sample means, standard deviations, and sample sizes to articulate how unlikely the observed sample difference is, assuming that the null hypothesis is true.

The significance level is a criterion used to decide whether to reject the null hypothesis. Commonly denoted by \( \alpha \), it quantifies the risk of incorrectly rejecting \( H_0 \). Typical significance levels are 0.05, 0.01, and 0.10.

In this exercise, we use a 0.05 significance level, which means there's a 5% risk of concluding there is a difference when there is none. It also defines the rejection region, dividing the probability distribution into a critical region: if the test statistic falls into this region, we reject \( H_0 \)

• At \( \alpha = 0.05 \), the critical t-value (two-tailed) provides boundaries for the decision rule.
• For degrees of freedom \( df = 7 \), the critical value is approximately \( \pm 2.3646 \).

With the calculated test statistic \( t = -1.38 \), we compare it against \( \pm 2.3646 \). Without reaching the critical value, we do not reject \( \mu_1 = \mu_2 \), implying insufficient evidence to declare a substantial difference.