Question

Home sellers participating in a survey indicated that it typically took between 6 and 14 months to sell a house. Suppose that the distribution of the time that it took to sell a house could be approximated by a normal distribution with a mean of 10 months and a standard deviation of 4 months.

(a) What percent of the homes sold took less than 9 months to sell?

(b) What percent of the homes sold took between 10 and 16 months to sell?

(c) How long did it take for 70% of homes to sell?

(d) If 25 sold homes randomly selected, what is the probability that the mean selling time of is less than 9 months?

(e) Is either of the following less likely than the other? If so, which one and why? If not, why not?

(i) Randomly selecting a home that takes less than 9 months to sell

(ii) Finding a random sample of 25 homes with a mean selling time of less than 9 months.

Answer 1

(a)

Let T be the time taken to sell a house. Then T ~ Normal( = 10, = 4)

Percent of the homes sold took less than 9 months to sell = P(T < 9)

= P[Z < (9 - 10) / 4]

= P[Z < -0.25]

= 0.4013 = 40.13%

(b)

Percent of the homes sold took less than between 10 and 16 months to sell = P(10 < T < 16)

= P[T < 16] - P[T < 10]

= P[Z < (16 - 10) / 4] - P[Z < (10 - 10) / 4]

= P[Z < 1.5] - P[Z < 0]

= 0.9332 - 0.5

= 0.4332 = 43.32%

(c)

Let t be the time taken for 70% of homes to sell.

P[T < t] = 0.70

P[Z < (t - 10)/4] = 0.70

(t - 10) / 4 = 0.5244 (Using Z distribution table)

t = 10 + 4 * 0.5244 = 12.0976 months

(d)

Standard error of mean = 4 / = 0.8

Then, mean selling time ~ Normal( = 10, = 0.8)

p=Probability that the mean selling time of is less than 9 months = P( < 9)

= P[Z < (9 - 10) / 0.8]

= P[Z < -1.25]

= 0.1056 = 10.56%

(e)

Finding a random sample of 25 homes with a mean selling time of less than 9 months is less likely than the Randomly selecting a home that takes less than 9 months to sell, because the standard deviation of mean selling time (0.8) is less than that of the random selling time or, the variation of mean selling time is less than that for random selling time.