Christmas Spending. In a recent study done by the National Retail Federation fou

Question

Christmas Spending. In a recent study done by the National Retail Federation found that 2017 Christmas spending for all US households follow a Normal distribution with mean $950

and a standard deviation $60. Use this information to answer the following questions.

1) Provide the z-score corresponding to the 2017 Christmas spending of $1200, and the probability of 2017 Christmas spending for a US household is more than $1200.

2) What is the proportion of 2017 Christmas spending within 1.30 standard deviations from the mean? Answer this question by completing parts 2(i). through (iii).

i. Provide the SMALLER z-score corresponding to the above statement.

ii. Provide the LARGER z-score corresponding to the above statement.

iii. Based on your answers in (i) and (ii), what proportion of 2017 Christmas spending is within 1.30 standard deviations from the mean?

Explanation / Answer

1) z-score = (x - mean)/sd = (1200 - 950)/60 = 4.17

P(x > 1200) = 1 - P(x < 1200)

                       = 1 - P((x - mean)/sd < ( 1200 - 950)/60)

                       = 1 - P(z < 4.17)

                      = 1 - 1 = 0

2) i) Smaller z - score = ((950 - 1.3 * 60) - 950)/60 = (872 - 950)/60 = -1.3

ii) Larger z-score = ((950 + 1.3 * 60) - 950)/60 = (1028 - 950)/60 = 1.3

iii) P(-1.3 < z < 1.3)

= P(Z < 1.3) - P(z < -1.3)

= 0.9032 - 0.0968

= 0.8064

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