SOLUTION: The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket.

Question 1084739: The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of dollar-sign 85 and a standard deviation of dollar-sign 25. If the management wants to give free gifts to at most 10.56% of the customers, what should the amount be above which a customer would receive a free gift?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the mean is 85
the standard deviation is 25.

you want only the top 10.56% of the customers to receive the gift.

those customers will be the ones who spend a certain amount more than the mean.

you need to find the z-score for an area to the right of that z-score equal to .1056.

subtract .1056 from 1 and you get .8944.

this is the area to the left of that same z-score.

look up in the z-score table, or use a z-score calculator, to find the z-score that has .8944 area under the normal distribution curve to the left of it.

i used a calculator and i got the z-score equal to 1.250272557.

to find the raw score from this z-score, use the z-score formula shown below:

z = (x - m) / s

in this problem:

z = 1.250272557
m = 85
s = 25

you will get 1.250272557 = (x - 85) / 25

solve for x to get x = 25 * 1.250272557 + 85 = 116.2568139

customers who spend more than 116.2568139 dollars will get the free gift.

the percentage of customers who get the gift will be at most 10.56%.

this can be seen visually by using the following online calculator:

http://davidmlane.com/hyperstat/z_table.html

you would enter the mean as 85 and the standard deviation as 25 and then click on above and enter 116.2568139.

the calculator will tell you that the area above that score is .1056 which is equal to 10.56%.

here's the display.

$$$

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