Questions: 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 13.3 % 25-36 34.9 % 37-48 31.0 % 49-60 20.2 % More than 60 0.6 % (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.

Solution Steps

The midpoints for each lease length category are calculated as follows:

• \(13-24\): midpoint = \(\frac{13 + 24}{2} = 18.5\)
• \(25-36\): midpoint = \(\frac{25 + 36}{2} = 30.5\)
• \(37-48\): midpoint = \(\frac{37 + 48}{2} = 42.5\)
• \(49-60\): midpoint = \(\frac{49 + 60}{2} = 54.5\)
• More than 60: midpoint = 66.5

The expected lease term is calculated using the formula: \[ E(X) = \sum (x_i \cdot p_i) \] where \(x_i\) is the midpoint and \(p_i\) is the probability (percent of leases).

\[ E(X) = 18.5 \times 0.133 + 30.5 \times 0.349 + 42.5 \times 0.310 + 54.5 \times 0.202 + 66.5 \times 0.006 \]

\[ E(X) = 2.4605 + 10.6445 + 13.175 + 11.009 + 0.399 \]

\[ E(X) = 37.688 \]

The standard deviation is calculated using the formula: \[ \sigma = \sqrt{\sum (x_i - E(X))^2 \cdot p_i} \]

\[ \sigma = \sqrt{(18.5 - 37.688)^2 \times 0.133 + (30.5 - 37.688)^2 \times 0.349 + (42.5 - 37.688)^2 \times 0.310 + (54.5 - 37.688)^2 \times 0.202 + (66.5 - 37.688)^2 \times 0.006} \]

\[ \sigma = \sqrt{(-19.188)^2 \times 0.133 + (-7.188)^2 \times 0.349 + (4.812)^2 \times 0.310 + (16.812)^2 \times 0.202 + (28.812)^2 \times 0.006} \]

\[ \sigma = \sqrt{49.168 + 17.996 + 7.181 + 57.196 + 4.978} \]

\[ \sigma = \sqrt{136.519} \]

\[ \sigma = 11.686 \]

Final Answer