Question

You may need to use the appropriate appenda table or technology to answer this queation. The data an y a annual sales ($1,000 s) for new customer accounts and x = number of vears of expenence for a sample of 10 salespersons brovided the etfimated regresion. equation y=80+4x . For these data, x=7,x(x¹−x² =142, and s=4.6098 : (a) Develop a 95%6 confidence interval far the mean annual sales (in thousands of dollars) for all salespersons with twelve years of eiperience (fiound your ansaars to two decimal places.) 5 thousand to $ theusand (b) The company is considering hiring Tom 5 mart, a salesperson with twelve years of expenience. Develop a 95% prediction inteival of annual sales (in thousands of dalars) for Tom Smart. (Round your answers to two decimal places.) $ thousand to $ thousand

Answer 1

Final answer:

The 95% confidence interval for the mean annual sales for all salespersons with twelve years of experience is approximately $104,704 to $111,296. The 95% prediction interval of annual sales for Tom Smart is approximately $116,477 to $139,523.

Step-by-step explanation:

To develop a 95% confidence interval for the mean annual sales for all salespersons with twelve years of experience, we can use the estimated regression equation y=80+4x. Since x=7, we can substitute this value into the equation to find the estimated mean annual sales for salespersons with twelve years of experience:

y = 80 + 4(7) = 80 + 28 = 108 (thousands of dollars)

Next, we need to calculate the standard error of the mean. The formula for the standard error of the mean is:

SE = s / sqrt(n)

Where s is the sample standard deviation and n is the sample size. In this case, s=4.6098 and n=10. Therefore, the standard error of the mean is:

SE = 4.6098 / sqrt(10) ≈ 1.457 (thousands of dollars)

To calculate the margin of error, we need to find the critical value for a 95% confidence level. Since the sample size is small (n<30), we can use the t-distribution. With a sample size of 10 and a 95% confidence level, the critical value is approximately 2.262.

The margin of error is then calculated as:

ME = critical value * SE = 2.262 * 1.457 ≈ 3.296 (thousands of dollars)

Finally, we can construct the confidence interval by subtracting and adding the margin of error to the estimated mean annual sales:

Confidence Interval = (Estimated Mean - Margin of Error, Estimated Mean + Margin of Error)

Confidence Interval = (108 - 3.296, 108 + 3.296) = (104.704, 111.296) (thousands of dollars)

Therefore, the 95% confidence interval for the mean annual sales for all salespersons with twelve years of experience is approximately $104,704 to $111,296.