Thank you for uploading the images. I see they relate to confidence interval calculations. Let's go step by step to find the correct values.

(a) Calculations

Given:

• $\bar{x} = 211.2$ (sample mean)
• $s = 44.37$ (sample standard deviation)
• $n = 27$ (sample size)
• Confidence level = 95%

1. Calculate the standard error (SE): $SE = \frac{s}{\sqrt{n}} = \frac{44.37}{\sqrt{27}} = \frac{44.37}{5.196} \approx 8.54$

2. Determine the critical $t$-value: For $n-1 = 26$ degrees of freedom and 95% confidence level, the $t$-value is approximately 2.056 (from $t$-table or software).

3. Margin of Error (ME): $ME = t \times SE = 2.056 \times 8.54 \approx 17.55$

4. Confidence interval: $\text{Lower limit} = \bar{x} - ME = 211.2 - 17.55 \approx 193.65$ $\text{Upper limit} = \bar{x} + ME = 211.2 + 17.55 \approx 228.75$

(b) Calculations

Given:

• $\bar{x} = 74.42$
• $s = 31.8$
• $n = 15$
• Confidence level = 99%

1. Calculate the standard error (SE): $SE = \frac{s}{\sqrt{n}} = \frac{31.8}{\sqrt{15}} = \frac{31.8}{3.873} \approx 8.21$

2. Determine the critical $t$-value: For $n-1 = 14$ degrees of freedom and 99% confidence level, the $t$-value is approximately 2.977.

3. Margin of Error (ME): $ME = t \times SE = 2.977 \times 8.21 \approx 24.44$

4. Confidence interval: $\text{Lower limit} = \bar{x} - ME = 74.42 - 24.44 \approx 49.98$ $\text{Upper limit} = \bar{x} + ME = 74.42 + 24.44 \approx 98.86$

Final Results

• (a) Confidence interval: $[193.65, 228.75]$
• (b) Confidence interval: $[49.98, 98.86]$

Let me know if you'd like further clarifications or step-by-step explanations!

Follow-Up Questions

1. How does changing the confidence level affect the interval width?
2. Why is the $t$-distribution used instead of the normal distribution in these cases?
3. What happens to the confidence interval when the sample size increases?
4. Can you explain the role of the critical $t$-value in the calculation?
5. How would the calculations differ if the population standard deviation were known?

Tip: Always ensure to check the degrees of freedom when finding the critical $t$-value for smaller sample sizes!