Question

A family is relocating from St. Louis, Missouri, to California. Due to an increasing inventory of houses in St. Louis, it is taking longer than before to sell a house. The wife is concerned and wants to know when it is optimal to put their house on the market. Her realtor friend informs them that the last 28 houses that sold in their neighborhood took an average time of 220 days to sell. The realtor also tells them that based on her prior experience, the population standard deviation is 40 days.

(A) What assumption regarding the population is necessary for making an interval estimate for the population mean? Assume that the central limit theorem applies. Assume that the population has a normal distribution.
(B) Construct the 90% confidence interval for the mean sale time for all homes in the neighborhood.

Answer 1

Explanation:

Hello!

The variable of interest is

X: Time it takes to sell a house in St. Lois

n= 28 houses

X[bar]= 220 days

σ= 40 days

a) To be able to apply the central limit theorem you have to use a large enough sample. As a rule, most statistic courses consider a sample size n≥30 to be large enough for the approximation.

You can always just assume that the population has a normal distribution if you had the raw information of the sample you could conduct a test to see if it comes from a normal population or not.

The sample taken is n=28, so in theory, you cannot apply the central limit theorem. Best is to assume the population is normal-

b) So assuming that the population has a normal distribution, the formula for the CI is:

X[bar] ±
`Z_(1-\alpha /2)` *
`(Sigma)/(√(n) )`

`Z_(1-\alpha /2)= Z_(0.95)= 1.648`

220 ± 1.648 *
`(40)/(√(28) )`

[207.54; 232.46]days

With a confidence level of 90%, you'd expect that the interval [207.54; 232.46]days will contain the true average time it takes to sell a house in St. Lois, Missouri.

I hope it helps!