QUESTION 4 Nationwide, the mean amount of sales each week at Lowe's store is normally distributed with a mean of \( \$ 5.8 \) million with a standard deviation of \( \$ 0.78 \) million. Lowe's has decided to close \( 15 \% \) of its stores, and has chosen amount of sales as the criterion on which the decision will be based (they are going to close the \( 15 \% \) of the stores with the lowest sales). How much in sales does a store have to have in order to not be closed? (please express your answer in millions and round your answer to 2 decimal places)

To find the sales threshold for stores that will not be closed, we need to determine the sales amount that corresponds to the 15th percentile of the normal distribution. Using the mean \( \mu = 5.8 \) million and standard deviation \( \sigma = 0.78 \) million, we lookup the z-score that corresponds to the 15th percentile, which is approximately \( z \approx -1.04 \). Now, we can use the z-score formula to find the sales amount \( X \): \[ X = \mu + z \cdot \sigma \] \[ X = 5.8 + (-1.04) \cdot 0.78 \approx 5.8 - 0.8112 \approx 4.99 \] Thus, a store must have sales of at least approximately **\$4.99 million** to avoid closure. --- Many businesses utilize statistical methods to make data-driven decisions about resource allocation and closures. For instance, using standard deviation and mean sales helps Lowe's determine which stores fall beneath expected sales performance. This practical application of statistics ensures that the business maintains its most profitable locations while making informed decisions based on actual data trends. If you’re keen on diving deeper into concepts of normal distributions and percentile calculations, consider exploring books on statistics like "The Elements of Statistical Learning" by Hastie, Tibshirani, and Friedman. This resource can provide not only a solid understanding of statistical concepts but also their applications in real-world scenarios, catering to both novices and seasoned practitioners.