Question 7.5: Jed Home Stores Credit Card Purchases Study Jed Home Stores ......

(a) Given \bar{x}= R170, s = R22 and n = 25 credit card customers.

The standard error of \bar{x} is estimated by \sigma_{\bar{x}}\approx \frac{s}{\sqrt{n} }=\frac{22}{\sqrt{25} } = R4.40

Since σ is unknown, the t-distribution must be used to find the t-limits for the 95% confidence level. From the t-table (Table 2, Appendix 1) the t-value associated with α = 0.025 and df = n − 1 = 25 − 1 = 24 is 2.064.

The estimated margin of error is 2.064 × 4.4 = R9.08.

Lower limit: 170 – 2.064(4.4) = 170 – 9.08 = 160.92

Upper limit: 170 + 2.064(4.4) = 170 + 9.08 = 179.08

The 95% confidence interval estimate for μ is given by R160.92 ≤ μ ≤ R179.08.

Management Interpretation

There is a 95% chance that the actual mean value of all credit card purchases at Jed Home Stores lies between R160.92 and R179.08.

(b) The increase in sample size to n = 46 will change the values of both the t-value and the estimated standard error of \bar{x}.

The t-limit is now 2.014, since α = 0.025 and df = n − 1 = 45, and the estimated standard error of \bar{x} is:

\sigma_{\bar{x}}\approx \frac{s}{\sqrt{n} }=\frac{22}{\sqrt{46} } = R3.24

The estimated margin of error is 2.014 × 3.24 = R6.53.

Lower limit: 170 – 2.014(3.24) = 170 – 6.53 = 163.47

Upper limit: 170 + 2.014(3.24) = 170 + 6.53 = 176.53

Thus the 95% confidence interval estimate for μ is given by R163.47 ≤ μ ≤ R176.53.

Management Interpretation

There is a 95% chance that the actual mean value of all credit card purchases at Jed Home Stores lies between R163.47 and R176.53.