Questions: Use the t-distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance α, and sample sizes n1 and n2. Assume that the samples are random and independent and the populations are normally distributed. Complete parts (a) and (b). Ha: μ1 ≠ μ2, α=0.02, n1=15, n2=8 (a) Find the critical value(s) assuming that the population variances are equal. (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)

Use the t-distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance α, and sample sizes n1 and n2. Assume that the samples are random and independent and the populations are normally distributed. Complete parts (a) and (b). Ha: μ1 ≠ μ2, α=0.02, n1=15, n2=8 (a) Find the critical value(s) assuming that the population variances are equal. (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)
Transcript text: Use the $t$-distribution table to find the critical value(s) for the indicated alternative hypotheses, level of significance $\alpha$, and sample sizes $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$. Assume that the samples are random and independent and the populations are normally distributed. Complete parts (a) and (b). \[ H_{a}: \mu_{1} \neq \mu_{2}, \alpha=0.02, n_{1}=15, n_{2}=8 \] (a) Find the critical value(s) assuming that the population variances are equal. $\square$ (Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as needed.)
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Solution

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Solution Steps

Step 1: Determine the Degrees of Freedom

Since variances are assumed equal, we use the pooled variance method to calculate degrees of freedom (df): $df = n_1 + n_2 - 2 = 15 + 8 - 2 = 21$.

Step 2: Find the Critical Value(s) from the t-Distribution Table

For a two-tailed test ($\mu_1 \neq \mu_2$), the critical values at significance level $\alpha = 0.02$ are found at $\alpha/2 = 0.01$ and $1-\alpha/2 = 0.99$, which are -2.518 and 2.518, respectively.

Final Answer: The critical values are -2.518 and 2.518.

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