Problem 1: Finding the 90% Confidence Interval for the Proportion

Given:

• $x = 30$ (number of laptops requiring extensive repairs)
• $n = 800$ (total laptops sampled)
• Confidence level = 90% ($z = 1.645$, from z-table for 90%)

Step 1: Calculate Sample Proportion ($\hat{p}$):

$\hat{p} = \frac{x}{n} = \frac{30}{800} = 0.0375$

Step 2: Calculate Standard Error ($SE$):

The formula for standard error is: $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ Substitute values: $SE = \sqrt{\frac{0.0375(1 - 0.0375)}{800}} = \sqrt{\frac{0.03609375}{800}} = \sqrt{0.00004512} \approx 0.00672$

Step 3: Calculate Margin of Error ($MOE$):

$MOE = z \cdot SE = 1.645 \cdot 0.00672 \approx 0.01105$

Step 4: Construct Confidence Interval (CI):

$CI = \hat{p} \pm MOE = 0.0375 \pm 0.01105$ $CI = (0.02645, 0.04855)$

Final Answer:

The 90% confidence interval for the proportion of laptops requiring extensive repairs is (0.026, 0.049).

Problem 2: Sample Size for Confidence Interval Width

Given:

• Confidence level = 97% ($z = 2.17$, from z-table for 97%)
• Desired margin of error ($E$) = $0.25/2 = 0.125$
• Standard deviation ($\sigma$) = 1.2 minutes

Step 1: Use Sample Size Formula:

The formula for sample size ($n$) is: $n = \left( \frac{z \cdot \sigma}{E} \right)^2$ Substitute values: $n = \left( \frac{2.17 \cdot 1.2}{0.125} \right)^2 = \left( \frac{2.604}{0.125} \right)^2 = (20.832)^2 \approx 433.98$

Final Answer:

The required sample size is approximately 434 students.

Do you want a detailed explanation for any step, or do you have additional questions?

Here are 5 related questions to consider:

1. How does the z-value affect the confidence interval or required sample size?
2. Can we use a t-distribution instead of z-distribution in Problem 1, and if so, when?
3. How would the calculations change if the population size was small?
4. Why is the desired margin of error halved in Problem 2?
5. What assumptions are required for these confidence interval calculations?

Tip: Always round the sample size $n$ up to the nearest whole number to ensure sufficient precision.