Question

U.S. consumers are increasingly viewing debit cards as a convenient substitute for cash and checks. The average amount spent annually on a debit card is $7,100 (Kiplinger’s, August 2007). Assume that the average amount spent on a debit card is normally distributed with a standard deviation of $500. [You may find it useful to reference the z table.]

a. A consumer advocate comments that the majority of consumers spend over $8,000 on a debit card. Find a flaw in this statement. (Round "z"value to 2 decimal places and final answer to 4 decimal places.)


b. Compute the 25th percentile of the amount spent on a debit card. (Round "z" value to 3 decimal places and final answer to 1 decimal place.)


c. Compute the 75th percentile of the amount spent on a debit card. (Round "z" value to 3 decimal places and final answer to 1 decimal place.)


d. What is the interquartile range of this distribution? (Round "z" value to 3 decimal places and final answer to 1 decimal place.)

Answer 1

a)

Here, μ = 7100, σ = 500 and x = 8000. We need to compute P(X >= 8000). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (8000 - 7100)/500 = 1.8

Therefore,
P(X >= 8000) = P(z <= (8000 - 7100)/500)
= P(z >= 1.8)
= 1 - 0.9641 = 0.0359

b)

z value at 25% = -0.674

z = (x - mean)/s

-0.674 = (x - 7100)/500
x = -0.674 * 500 + 7100
x = 6763.0

c)

z value at 75% = 0.674

z = (x - mean)/s

0.674 = (x - 7100)/500
x = 0.674 * 500 + 7100
x = 7437

d)

IQR = Q3 - Q1
= 7437 - 6763
= 674