To solve this problem, we need to construct a 95% confidence interval for the proportion of all digital camera customers who purchased the new warranty. The steps involved are as follows:

Step 1: Define the sample proportion

The sample proportion $\hat{p}$ is given by:

$\hat{p} = \frac{x}{n}$

Where:

• $x = 169$ (the number of customers who purchased the new warranty)
• $n = 535$ (the total number of customers in the sample)

$\hat{p} = \frac{169}{535} \approx 0.316$

Step 2: Find the standard error of the proportion

The standard error (SE) for the proportion is calculated using the formula:

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Substituting the values:

$SE = \sqrt{\frac{0.316(1 - 0.316)}{535}} = \sqrt{\frac{0.316 \times 0.684}{535}} \approx 0.020$

Step 3: Find the critical value for a 95% confidence interval

For a 95% confidence interval, the critical value (z-value) from the standard normal distribution is 1.96.

Step 4: Calculate the margin of error

The margin of error (ME) is given by:

$ME = z \times SE = 1.96 \times 0.020 \approx 0.039$

Step 5: Construct the confidence interval

The 95% confidence interval for the proportion is:

$\hat{p} \pm ME = 0.316 \pm 0.039$

Thus, the confidence interval is:

$(0.316 - 0.039, 0.316 + 0.039) = (0.277, 0.355)$

Step 6: Interpret the result

We are 95% confident that the true proportion of all digital camera customers who have purchased the new warranty is between 0.277 and 0.355.

Step 7: Compare with 0.20

Since the lower bound of the confidence interval (0.277) is greater than 0.20, we are 95% confident that the proportion of customers who purchased the new warranty is greater than 0.20.

Conclusion:

Yes, we are 95% confident that the proportion of customers who purchased the new warranty exceeds 0.20.

Would you like any more details on the process or have any questions?

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Here are 5 related questions you might consider:

1. What is the formula for a confidence interval for proportions, and why is it used?
2. How would the confidence interval change if we used a different confidence level (e.g., 90%)?
3. What is the impact of sample size on the width of a confidence interval?
4. How can we test whether the increase in warranty purchases is statistically significant?
5. What assumptions do we make when constructing a confidence interval for proportions?

Tip: Increasing the sample size reduces the margin of error, which leads to a narrower confidence interval.