Question

A company claims that at least 70% of its customers are satisfied with their products. A survey of 500 customers is conducted, and 345 of them indicate satisfaction.

a. Find the 99% confidence interval for the true proportion of satisfied customers.

b. Based on the confidence interval you just made, can you say the company’s estimate of 70% satisfaction is wrong Why?

c. Based on the confidence interval you just made, can you say the company’s estimate of 70% satisfaction is correct? Why?

Answer 1

The 99% confidence interval for the true proportion of satisfied customers is approximately 65.2% to 73.8%.

Step-by-step explanation:

To find the 99% confidence interval for the true proportion of satisfied customers, we can use the formula for the confidence interval of a proportion:

`\[ \text{Confidence Interval} = \hat{p} \pm Z * \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]`

where:

`- \(\hat{p}\)` is the sample proportion (345/500 = 0.69),

`- \(Z\)` is the critical value for a 99% confidence level (approximately 2.576 for a large sample size),

`- \(n\)` is the sample size (500).

Substitute these values into the formula:

`\[ \text{Confidence Interval} = 0.69 \pm 2.576 * \sqrt{(0.69 * (1-0.69))/(500)} \]`

After calculation, the confidence interval is approximately 65.2% to 73.8%.

The company's claim of at least 70% satisfaction falls within this interval, suggesting that the estimate is plausible. However, it's important to note that the interval does not include exactly 70%, indicating that the claim might be slightly optimistic. The confidence interval provides a range of plausible values for the true proportion, and in this case, it does not rule out the possibility that the true satisfaction rate is lower than the company's claim.