Question

To determine the optimal number of items a company should hold in inventory, it is necessary to estimate the average weekly sales of that particular item. A random sample is taken, and the number of items sold per week is as follows: 64, 57, 49, 81, 76, 70, 59. Assume that the number of items sold is normally distributed. Give a point estimate and interval estimates for the mean number of items sold per week, assuming a 80%, 90%, 95% and 99% confidence level. Explain the meaning of your interval estimates.

Answer 1

$\dpi{100} \bar x$ = (64 + 57 + 49 + 81 + 76 + 70 + 59)/7 = 65.143

s = sqrt(((64 - 65.143)^2 + (57 - 65.143)^2 + (49 - 65.143)^2 + (81 - 65.143)^2 + (76 - 65.143)^2 + (70 - 65.143)^2 + (59 - 65.143)^2)/6) = 11.2462

At 80% confidence interval the critical value is t^* = 1.440

The 80% confidence interval for population mean is

$\dpi{100} \bar x$ +/- t^* * s/$\dpi{100} \sqrt n$

= 65.143 +/- 1.440 * 11.2462/$\dpi{100} \sqrt 7$

= 65.143 +/- 6.121

= 59.022, 71.264

We are 80% confident that the true population mean number of items sold per week lies in the above interval

At 90% confidence interval the critical value is t^* = 1.943

The 90% confidence interval for population mean is

$\dpi{100} \bar x$ +/- t^* * s/$\dpi{100} \sqrt n$

= 65.143 +/- 1.943 * 11.2462/$\dpi{100} \sqrt 7$

= 65.143 +/- 8.259

= 56.884, 73.402

We are 90% confident that the true population mean number of items sold per week lies in the above interval

At 95% confidence interval the critical value is t^* = 2.447

The 95% confidence interval for population mean is

$\dpi{100} \bar x$ +/- t^* * s/$\dpi{100} \sqrt n$

= 65.143 +/- 2.447 * 11.2462/$\dpi{100} \sqrt 7$

= 65.143 +/- 10.401

= 54.742, 75.544

We are 95% confident that the true population mean number of items sold per week lies in the above interval

At 99% confidence interval the critical value is t^* = 3.708

The 99% confidence interval for population mean is

$\dpi{100} \bar x$ +/- t^* * s/$\dpi{100} \sqrt n$

= 65.143 +/- 3.708 * 11.2462/$\dpi{100} \sqrt 7$

= 65.143 +/- 15.761

= 49.382, 80.904

We are 99% confident that the true population mean number of items sold per week lies in the above interval