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	14.8% 36.4% 29.5% 18.6% 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	months
standard deviation

(b) Sketch a graph of the probability distribution for the duration of new auto leases.

Answer 1

Solution:
Midpoint of each class can be calculated as
Midpoint = (Lower class limit+Upper class limit)/2

Lease Length in Months	P(% of Leases)	Midpoints of Each classes(M)
13-24	0.148	18.5
25-36	0.364	30.5
37-48	0.295	42.5
49-60	0.186	54.5
More than 60	0.007	66.5

Expected Lease term can be calculated as
Expected lease = (Mi * Pi) = (0.148*18.5) + (0.364*30.5) + (0.295*42.5) + (0.186*54.5) + (0.007*66.5)

Lease Length in Months	% of Leases	Midpoints of Each classes	P*M
13-24	0.148	18.5	2.738
25-36	0.364	30.5	11.102
37-48	0.295	42.5	12.5375
49-60	0.186	54.5	10.137
More than 60	0.007	66.5	0.4655

Expected Lease(mean) = (2.738 + 11.102+12.5375+10.137+0.4655) = 36.98
Standard deviation can be calculated as
Standard deviation = sqrt(((Mi - Mean)^2 * Pi)) = sqrt(((18.5-36.98)^2 *0.148 + (30.5-36.98)^2 * 0.364 + (42.5-36.98)^2 * 0.295 + (54.5-36.98)^2 * 0.186 + (66.5-36.98)^2 * 0.007))

Lease Length in Months	% of Leases	Midpoints of Each classes	P*M	(M-mean)	(M-mean)^2	(M-mean)^2 * P
13-24	0.148	18.5	2.738	-18.48	341.5104	50.54354
25-36	0.364	30.5	11.102	-6.48	41.9904	15.28451
37-48	0.295	42.5	12.5375	5.52	30.4704	8.988768
49-60	0.186	54.5	10.137	17.52	306.9504	57.09277
More than 60	0.007	66.5	0.4655	29.52	871.4304	6.100013

Standard deviation = sqrt(50.54354+15.28451+8.988768+57.09277+6.100013) = sqrt(138.0096) = 11.75
Solution(b)