Question

Consumer Banker Association released a report showing the lengths of automobile leases for new automobiles. The results are as follows.

Lease Length in Months	Percent of Leases
13-24 25-36 37-48 49-60 More than 60	15.4% 36.4% 25.6% 21.9% 0.7%

(a) Use the midpoint of each class, and call the midpoint of the last class 66.5 months, for purposes of computing the expected lease term. Also find the standard deviation of the distribution. (Round your answers to two decimal places.)

expected lease term
standard deviation

Answer 1

Solution:

The formula for mean and standard deviation for the probability distribution is given as below:

Mean = ∑XP(X)

Variance = ∑ P(X)*(X - mean)^2

SD = sqrt[∑ P(X)*(X - mean)^2]

The calculation table is given as below:

Class	Midpoint X	P(X)	XP(X)	(X - Mean)^2	P(X)*(X - mean)^2
13 to 24	18.5	0.154	2.849	350.887824	54.0367249
25 to 36	30.5	0.364	11.102	45.319824	16.49641594
37 to 48	42.5	0.256	10.88	27.751824	7.104466944
49 to 60	54.5	0.219	11.9355	298.183824	65.30225746
More than 60	66.5	0.007	0.4655	856.615824	5.996310768
	Total	1	37.232		148.936176

From above table, we have

Mean = ∑XP(X) = 37.232

Expected lease term = Mean = 37.232

Variance = ∑ P(X)*(X - mean)^2

Variance = 148.936176

SD = sqrt(148.936176)

Standard deviation = 12.203941