Let’s go through these two questions one by one. I'll break down the solution process for each part:

Question 34

1. Given data:

   • Sample size $n = 50$
   • Sample mean $\bar{x} = 26$ weeks
   • Sample standard deviation $s = 6.2$ weeks
   • Confidence level = 95%

2. Constructing a 95% confidence interval for the population mean:

   • Since the sample size is 50, which is sufficiently large, we can use the normal distribution for the confidence interval.
   • For a 95% confidence level, the z-score is approximately 1.96.

   The formula for the confidence interval is: $\text{Confidence Interval} = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}}$ Substituting the values: $= 26 \pm 1.96 \cdot \frac{6.2}{\sqrt{50}}$

3. Interpretation:

   • After calculating, check if the interval includes 28 weeks to determine if it is reasonable for the population mean to be 28 weeks.

Question 35

Given data:

• Sample size $n = 40$
• Sample mean $\bar{x} = 8.32$ years
• Sample standard deviation $s = 3.07$ years
• Confidence level = 90%

Solving each part:

• (a) Mean of the population: The sample mean $\bar{x}$ of 8.32 years is an estimate of the population mean.

• (b) Constructing a 90% confidence interval for the population mean:

  • With $n = 40$, we use the t-distribution as a more accurate method for smaller samples. The t-value for a 90% confidence level with $n - 1 = 39$ degrees of freedom is approximately 1.685 (from the t-table).

  The confidence interval formula is: $\text{Confidence Interval} = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}}$ Substituting the values: $= 8.32 \pm 1.685 \cdot \frac{3.07}{\sqrt{40}}$

• (c) Verifying the previous director's claim of “almost 10 years”:

  • Check if the interval from part (b) includes 10 years. If it does, it supports the claim; otherwise, it does not.

Would you like me to go through the calculations in detail for each step?

Here are some related questions you might consider:

1. How does the sample size affect the width of a confidence interval?
2. Why is the t-distribution used for smaller sample sizes instead of the normal distribution?
3. How would changing the confidence level to 99% affect the interval for each question?
4. What assumptions must be met to construct a valid confidence interval?
5. How does the standard deviation impact the confidence interval range?

Tip: Confidence intervals provide a range of values within which we expect the true population parameter to lie, given a certain level of confidence.