Chapter 8: Problem 15

Sales personnel for Skillings Distributors submit weekly reports listing the customer contacts made during the week. A sample of 65 weekly reports showed a sample mean of 19.5 customer contacts per week. The sample standard deviation was \(5.2 .\) Provide \(90 \%\) and \(95 \%\) confidence intervals for the population mean number of weekly customer contacts for the sales personnel.

Short Answer

Expert verified

The 90% confidence interval for the population mean number of weekly customer contacts for sales personnel is approximately (18.4216, 20.5784), and the 95% confidence interval is approximately (18.2085, 20.7915).

Step by step solution

Identify given values

In this exercise, we are given the following values: - Sample size (n) = 65 - Sample mean (x̅) = 19.5 - Sample standard deviation (s) = 5.2

Calculate the standard error

The standard error (SE) is the standard deviation divided by the square root of the sample size. We will use this formula to calculate the standard error: SE = s / sqrt(n) Plugging in the given values: SE = 5.2 / sqrt(65) ≈ 0.6464

Find the critical values for 90% and 95% confidence intervals

To find the critical values, we can use t-distribution since the population standard deviation is unknown. We will use the degrees of freedom (df), which are equal to the sample size minus 1: df = n - 1 = 65 - 1 = 64 For a 90% confidence level, the t-score (t90) is found using a t-distribution table or an online calculator. We find that: t90 ≈ 1.667 Similarly, for a 95% confidence level, the t-score (t95) is: t95 ≈ 1.998

Calculate the margin of error for 90% and 95% confidence intervals

We will use the formula to calculate the margin of error for both confidence intervals: Margin of error = t * SE For 90% confidence interval: Margin of error90 = t90 * SE = 1.667 * 0.6464 ≈ 1.0784 For 95% confidence interval: Margin of error95 = t95 * SE = 1.998 * 0.6464 ≈ 1.2915

Calculate the confidence intervals

Finally, we will calculate the confidence intervals using the sample mean and the margin of error: Confidence interval = (x̅ - margin of error, x̅ + margin of error) For 90% confidence interval: Confidence interval90 = (19.5 - 1.0784, 19.5 + 1.0784) = (18.4216, 20.5784) For 95% confidence interval: Confidence interval95 = (19.5 - 1.2915, 19.5 + 1.2915) = (18.2085, 20.7915) In conclusion, we can be 90% confident that the true population mean number of weekly customer contacts for sales personnel lies between 18.4216 and 20.5784, and 95% confident that it lies between 18.2085 and 20.7915.

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