Questions: Independent random samples from normal populations produced the results shown in the table to the right. Assume that the population variances are equal. Complete parts a through d below. a. Calculate the pooled estimate of σ². sᵖ²=0.5731 (Round to four decimal places as needed.) b. Do the data provide sufficient evidence to indicate that μ₂>μ₁? Test using α=0.10. What are the null and alternative hypotheses? A. H₀ · μ₁-μ₂=0.10 B. H₀: μ₁-μ₂=0 Hᵃ: μ₁-μ₂ ≤ 0.10 Hᵃ · μ₁-μ₂<0 C. H₀: μ₁-μ₂=0.10 D. H₀: μ₁-μ₂=0 Hᵃ: μ₁-μ₂>0 What is the test statistic? t= (Round to two decimal places as needed.)

Independent random samples from normal populations produced the results shown in the table to the right. Assume that the population variances are equal. Complete parts a through d below. a. Calculate the pooled estimate of σ². sᵖ²=0.5731 (Round to four decimal places as needed.) b. Do the data provide sufficient evidence to indicate that μ₂>μ₁? Test using α=0.10. What are the null and alternative hypotheses? A. H₀ · μ₁-μ₂=0.10 B. H₀: μ₁-μ₂=0 Hᵃ: μ₁-μ₂ ≤ 0.10 Hᵃ · μ₁-μ₂<0 C. H₀: μ₁-μ₂=0.10 D. H₀: μ₁-μ₂=0 Hᵃ: μ₁-μ₂>0 What is the test statistic? t= (Round to two decimal places as needed.)
Transcript text: Independent random samples from normal populations produced the results shown in the table to the right. Assume that the population variances are equal. Complete parts a through d below. a. Calculate the pooled estimate of $\sigma^{2}$. \[ s_{p}^{2}=0.5731 \] (Round to four decimal places as needed.) b. Do the data provide sufficient evidence to indicate that $\mu_{2}>\mu_{1}$ ? Test using $\alpha=0.10$. What are the null and alternative hypotheses? A. $H_{0} \cdot \mu_{1}-\mu_{2}=0.10$ B. $H_{0}: \mu_{1}-\mu_{2}=0$ $H_{a}: \mu_{1}-\mu_{2} \leq 0.10$ $H_{a} \cdot \mu_{1}-\mu_{2}<0$ C. $H_{0}: \mu_{1}-\mu_{2}=0.10$ D. \[ \begin{array}{l} H_{0}: \mu_{1}-\mu_{2}=0 \\ H_{a}: \mu_{1}-\mu_{2}>0 \end{array} \] What is the test statistic? $\mathrm{t}=$ $\square$ (Round to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the pooled standard deviation (Sp)

Using the formula $S_p = \sqrt{\frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1 + n_2 - 2}}$, we find $S_p = 0.76$.

Step 2: Compute the t-statistic (t_STAT)

Using the formula $t_{STAT} = \frac{\bar{X}_1 - \bar{X}_2}{S_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$, we find $t_{STAT} = -2.1$.

Step 3: Determine the degrees of freedom (df)

The degrees of freedom for the test is $df = n_1 + n_2 - 2 = 6$.

Step 4: Find the critical value(s) for the t-distribution

For a significance level of 0.1, the critical value(s) is/are -1.44.

Step 5: Make a decision

Since the computed $t_{STAT}$ is outside the range of the critical value(s), we reject the null hypothesis $H_0$.

Final Answer:

Based on the computed t-statistic and the critical value(s), we reject the null hypothesis $H_0$. This suggests that there is a significant difference between the two sample means at the given significance level.

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