Question

1. In order to get more female customers, a new clothing store offers free gourmet coffee and pastry to its customers. The average daily revenue over the past five-week period has been $1,080 with a standard deviation of $260. Use this sample information to construct a 95% confidence interval for the average daily revenue. The store manager believes that the coffee and pastry strategy would lead to an average daily revenue of $1,200. Is the manager correct based on the 95% confidence interval?

Answer 1

Answer:

No, the manager is not correct based on the 95% confidence interval.

Step-by-step explanation:

We are given that the average daily revenue over the past five-week period has been $1,080 with a standard deviation of $260, i.e.; X bar = $1080 and s = $260 and sample size, n = 35 .

The Pivotal quantity for 95% confidence interval is given by;

                $\frac{Xbar - \mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, X bar = sample mean = $1080

                s  = sample standard deviation = $260

                 n = sample size = 35 {five-week}

So, 95% confidence interval for average daily revenue, $\mu$ is given by;

P(-2.032 < $t_3_4$ < 2.032) = 0.95

P(-2.032 < $\frac{Xbar - \mu}{\frac{s}{\sqrt{n} } }$ < 2.032) = 0.95

P(-2.032 * ${\frac{s}{\sqrt{n} }$ < ${Xbar - \mu}$ < 2.032 * ${\frac{s}{\sqrt{n} }$ ) = 0.95

P(X bar - 2.032 * ${\frac{s}{\sqrt{n} }$ < $\mu$ < X bar + 2.032 * ${\frac{s}{\sqrt{n} }$ ) = 0.95

95% confidence interval for $\mu$ = [ X bar - 2.032 * ${\frac{s}{\sqrt{n} }$ , X bar + 2.032 * ${\frac{s}{\sqrt{n} }$ ]

                                            = [ 1080 - 2.032 * ${\frac{260}{\sqrt{35} }$ , 1080 + 2.032 * ${\frac{260}{\sqrt{35} }$ ]

                                             = [ 990.70 , 1169.30 ]

No, the manager is not correct based on the fact that the coffee and pastry strategy would lead to an average daily revenue of $1,200 because the calculate 95% confidence interval does not include value of $1200.

Therefore, the store manager believe is not correct.