Simple Regression Models Case Study: Mystery Shoppers Chic Sales is a high-end c

Question

Simple Regression Models Case Study: Mystery Shoppers Chic Sales is a high-end consignment store with several locations in the metro area. The company noticed a decrease in sales over the last fiscal year. Research indicated customer satisfaction had decreased and the owner, Pat Turner, decided to create a mystery shopper program. The mystery shopper program lasted over a 6-month period, employing several loyal and new customers assigned to each location. Surveys were on a 100-point scale and involved categories such as “Staff Attitude,” “Store Cleanliness,” “Product Availability,” and “Display(s) Appeal.” After the mystery shopper period concludes, Mrs. Turner sends you the following e-mail: From: Pat Turner Sent: Thursday, July 7, 2016 8:57 a.m. Subject: Mystery Data Shopper Stats and Store Performance? Good morning! Welcome back from vacation I hope you had a wonderful Fourth of July. The last mystery shopper surveys came in and I have the final numbers. I am interested in whether there is a way to predict the final average based on the initial survey score. Also, is there a statistically significant relationship between how stores initially performed and what the overall average is? The initial survey score and the final average data for all seven store locations is in the table below: Store 1 2 3 4 5 6 7 Initial Survey Score 83 97 84 72 85 64 93 Final Average 78 98 92 75 88 70 93 Also, how good is the relationship between Initial Survey Score and the Final Average? Could I use an Initial Survey Score to predict a Final Average? In fact, could I predict a Final Average if I have an Initial Survey Score of 90?

Explanation / Answer

The table for Initial Average score and Final Average score. The regression table

To calculate the strength of relationship, we will caclulate the correlation coefficient r.

r = [n(xy) - (x)((y)] / sqrt [ (n (x2 ) - (x)2) ( n (y2 ) - (y)2]

r = [ 7 * 49717 - 578 * 594 ] / sqrt [(7 * 48508 - 5782 ) * (7 * 51070 - 5942 )]

r = 4587 / sqrt (5472 * 4654) = 4587/ 5046.453 = 0.909

so strength of relation is quite strong as it is above 0.6 so relationship is linear and having strong strength.

Now linear regression equation

y = a + bx

where y = final average and x = initial average score

here a =  [(y) (x2 ) - (x) (xy)]/ [ n (x2 ) - (x)2 ]

a = [ 594 * 48508 - 578 * 49717] / [ 7 * 48508 - 5782 ]

a = 77326 / 5472 = 14.13

b =  [ n(xy) - (x)((y)]/ [ n (x2 ) - (x)2 ]

b = [ 7 * 49717 - 578 * 594] / [ 7 * 48508 - 5782 ]

b = 4687 / 5472 = 0.8565

so regresstion line y = 0.8565x + 14.131

so for initial survey score = 90

Final score = 0.8565 * 90 + 14.131 = 91.22

Store Initial Average(x) Final Average(y) x^2 xy y^2 1 83 78 6889 6474 6084 2 97 98 9409 9506 9604 3 84 92 7056 7728 8464 4 72 75 5184 5400 5625 5 85 88 7225 7480 7744 6 64 70 4096 4480 4900 7 93 93 8649 8649 8649 sum 578 594 48508 49717 51070

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