Question

Housing prices in Athens have been researched extensively by faculty at UGA. The current thinking is that housing prices follow an approximately normal model with mean $162,000 and standard deviation $7,034.

(a) What proportion of housing prices in Athens are less than $153,000? (3 decimal places)

(b) A realtor takes a random sample of 128 houses in Athens. Determine the probability the average selling price is greater than $163,000? (3 decimal places)

(c) A realtor in Asheville, NC wants to estimate the mean housing price of houses in Asheville. The realtor believes the distribution of housing prices in Asheville is similar to those in Athens.

If this realtor takes a random sample of 128 homes in Asheville, what is the standard error of the estimate? (3 decimal places) How many homes in Asheville should the realtor sample to be 98% confident the estimate is within $500 of the true mean price? Use the critical value to exactly 3 decimal places.

Answer 1

Mean = = $ 162,000

standard deviation = = $ 7034

(a) P(x < $ 153,000)

z = (153000 -162000)/7034 = -1.2795

P(x < $ 153,000) = P(Z < -1.2795) = 0.1004

(b) n = 128

standard error = /sqrt(n) = 7034/sqrt(128) = $ 621.72

P(x > $ 163,000)

Z = (163000 - 162000)/621.72 = 1.6084

P(x > $ 163,000) = 1- P(Z < 1.6084)

= 1- 0.9461 = 0.0539

(c) standard error of the estimate = /sqrt(n) = 7034/sqrt(128) = $ 621.72

margin of error = $ 500

Critical test statistic for 98% confidence interval = 2.326

so,

500 = 2.326 * 7034/sqrt(n)

n = 1071