Question

The distribution of online sale price for four-year-old Harley-Davidson touring motorcycles is approximately Normally distributed with a mean of $15,000 and a standard deviation of $4,000.

(A) Mr. Rampal plans to spend between $9,000 and $12,000 on one of these motorcycles. What proportion of the available motorcycles of this type can he afford?

(B)What is the 30th percentile for the prices of motorcycles of this type?

(C)Show that a motorcycle of this type priced at $28,000 is considered an outlier by the 1.5xIQR rule.

Answer 1

(A)

Given,

= $15,000

= $4,000,

Proportion of the available motorcycles of this type can he afford = P($9,000 X $12,000)

= P[(9000 - 15000)/4000 Z (12000 - 15000)/4000 ]

= P(-1.5 Z -0.75)

= P(Z -0.75) - P(Z -1.5)

= 0.2266 - 0.0668

= 0.1598 (Using standard normal tables)

(B)

For 30th percentile,

P(X < x) = 0.3

P[Z < (x - 15000)/4000] = 0.3

(x - 15000)/4000 = -0.5244     (Using standard normal tables)

=> x = 15000 - 0.5244 * 4000 = $12902.4

30th percentile for the prices of motorcycles of this type is $12902.4

(C)

For 25th percentile,

P(X < x) = 0.25

P[Z < (x - 15000)/4000] = 0.25

(x - 15000)/4000 = -0.6745     (Using standard normal tables)

=> x = 15000 - 0.6745 * 4000 = $12302

25th percentile (Q1) for the prices of motorcycles of this type is $12302

For 75th percentile,

P(X < x) = 0.75

P[Z < (x - 15000)/4000] = 0.75

(x - 15000)/4000 = 0.6745    (Using standard normal tables)

=> x = 15000 + 0.6745 * 4000 = $17698

75th percentile (Q3) for the prices of motorcycles of this type is $17698

IQR = Q3 - Q1 = 17698 - 12302 = 5396

Q3 + 1.5xIQR = 17698 + 1.5 * 5396 = 25792

Since, the price of 28,000 is greater than Q3 + 1.5xIQR, a motorcycle of this type priced at $28,000 is considered an outlier.