Question

Normal
mu	722
sigma	189
xi	P(X<=xi)
151	0.0013
263	0.0076
532	0.1574
721	0.4979
810	0.6793
961	0.8970
P(X<=xi)	xi
0.11	490.1862
0.12	499.9275
0.24	588.5088
0.31	628.2843
0.38	664.2641
0.76	855.4912
0.89	953.8138

Use the cumulative normal probability excel output above (dealing with the amount of money parents spend per child on back-to-school items) to answer the following question.

The probability is 0.38 that the amount spent on a randomly selected child will be between two values (in $) equidistant from the mean. The lower of these equidistant points is provided in the excel output above. Use the lower endpoint and some math to find the upper endpoint.

Answer 1

Here the probability is 0.38 that the the amount spent on a randomly selectted child will be between two values (in $) equidistant from the mean.

that mean the probability values are

for lower endpoint = 0.50 - 0.38/2 = 0.31

for upper endpoint = 0.50 + 0.38/2 = 0.69

here for 0.31, the lower end point is = 628.2843

Now as we know that Z value for lower end point is

Z = -0.49585 = (628. 2843 - )/ ...(i)

now to find the value of normal distribution parameter

we will take one more value let say

P(X < = xi) = 0.76 for x = 855.4912

so Z value = 0.7063 = (855.4912 - )/ ...(ii)

now doing (ii)/(i)

0.7063/0.49585 = ( - 855.4912) / (628. 2843 - )

1.42443 *  (628. 2843 - ) = ( - 855.4912)

2.42443 = 1750.436

= 722

=  (855.4912 - )/0.7063 = 189

so here

for cumulative distribuition function F(Z) = 0.69

so here Z value for the given is = 0.49585

so here

upper endpoint = 722 + 0.49585 * 189 = 815.7157

Lower endpoint = 628.2843

Upper endpoint = 815.7157