Questions: Module 4 Chapter 7 (Part A) Module 4: Chapter 7 (Part A) Pr σ2=11.03 (a) What distribution (standard normal or Student's t) should be used to construct a confidence interval for μ1-μ2? Explain. A standard normal should be used because σ1 and σ2 are known. A Student's t should be used because σ1 and σ2 are unknown. A Student's t should be used because σ1 and σ2 are known. A standard normal should be used because σ1 and σ2 are unknown. Because the interval contains only negative numbers, we can say that the fathers have a higher mean empathy score. Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score. Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score. We can not make any conclusions using this confidence interval.

Module 4 Chapter 7 (Part A) Module 4: Chapter 7 (Part A) Pr σ2=11.03 (a) What distribution (standard normal or Student's t) should be used to construct a confidence interval for μ1-μ2? Explain. A standard normal should be used because σ1 and σ2 are known. A Student's t should be used because σ1 and σ2 are unknown. A Student's t should be used because σ1 and σ2 are known. A standard normal should be used because σ1 and σ2 are unknown. Because the interval contains only negative numbers, we can say that the fathers have a higher mean empathy score. Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score. Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score. We can not make any conclusions using this confidence interval.
Transcript text: Module $4 \mid$ Chapter 7 (Part A) \ Module 4: Chapter 7 (Part A) Pr \ $\sigma_{2}=11.03$ \ (a) What distribution (standard normal or Student's $t$ ) should be used to construct a confidence interval for $\mu_{1}-\mu_{2}$ ? Explain. \ A standard normal should be used because $\sigma_{1}$ and $\sigma_{2}$ are known. \ A Student's $t$ should be used because $\sigma_{1}$ and $\sigma_{2}$ are unknown. \ A Student's $t$ should be used because $\sigma_{1}$ and $\sigma_{2}$ are known. \ A standard normal should be used because $\sigma_{1}$ and $\sigma_{2}$ are unknown. \ Because the interval contains only negative numbers, we can say that the fathers have a higher mean empathy score. \ Because the interval contains only positive numbers, we can say that the mothers have a higher mean empathy score. \ Because the interval contains both positive and negative numbers, we can not say that the mothers have a higher mean empathy score. \ We can not make any conclusions using this confidence interval.
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Solution

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Solution Steps

Step 1: Determine the Distribution

To construct a confidence interval for the difference between two population means \( \mu_1 - \mu_2 \), we first assess the known variances. Since both \( \sigma_1 \) and \( \sigma_2 \) are known, we use the standard normal distribution.

Thus, the distribution to use is: \[ \text{Distribution to use: } \text{standard normal} \]

Step 2: Calculate the Confidence Interval

We calculate the confidence interval for the difference in means using the formula: \[ (\bar{x}_1 - \bar{x}_2) \pm z \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \] Substituting the values:

  • \( \bar{x}_1 = 50 \)
  • \( \bar{x}_2 = 45 \)
  • \( \sigma_1 = 10 \)
  • \( \sigma_2 = 11.03 \)
  • \( n_1 = 30 \)
  • \( n_2 = 30 \)
  • \( z \) for a 99% confidence level is approximately \( 2.58 \)

The calculation proceeds as follows: \[ 50 - 45 \pm 2.58 \cdot \sqrt{\frac{10^2}{30} + \frac{11.03^2}{30}} \] Calculating the standard error: \[ \sqrt{\frac{100}{30} + \frac{121.606409}{30}} = \sqrt{3.3333 + 4.0535} = \sqrt{7.3868} \approx 2.718 \] Thus, the confidence interval becomes: \[ 5 \pm 2.58 \cdot 2.718 \approx 5 \pm 7.014 \] This results in: \[ (-2.0, 12.0) \]

Step 3: Draw Conclusions from the Confidence Interval

The confidence interval for the difference in means is: \[ \text{Confidence Interval: } (-2.0, 12.0) \] Since this interval contains both negative and positive values, we conclude that: \[ \text{Conclusion: } \text{we cannot say that the mothers have a higher mean empathy score.} \]

Final Answer

The answer is: \[ \boxed{\text{Distribution: standard normal, Confidence Interval: (-2.0, 12.0), Conclusion: we cannot say that the mothers have a higher mean empathy score.}} \]

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