Question

In: Statistics and Probability

A report announced that the mean sales price of all new houses sold one year was...

A report announced that the mean sales price of all new houses sold one year was $272,000. Assume that the population standard deviation of the prices is $100,000. If you select a random sample of 100 new houses, what is the probability that the sample mean sales price will be between $250,000 and $285,000?

Select one:

a. 0.8034

b. 0.1388

c. 0.2956

d. 0.8893

Solutions

Expert Solution

Solution :

Let X be a population which represents the sale prices of new houses sold.

Mean (μ) = $272,000

SD (σ) = $100,000

Sample size (n) = 100

According to central limit theorem, if we have a population with mean μ and standard deviation σ and if we take a same of size n (n > 30), then sampling distribution of sample means of all such samples of this follows approximately normal distribution with mean μ and standard deviation σ/√n.

i.e.  x̅ ~ N(μ, σ²/n).

We have to find P($250,000 < x̅ < $285,000).

P($250,000 < x̅ < $285,000) = P(x̅ < $285,000) - P(x̅ ≤ $250,000)

We know that, if x̅ ~ N(μ, σ²/n) then,

Using "pnorm" function of R we get,

P(Z < 1.3) = 0.9032 and P(Z ≤ -2.2) = 0.0139

Hence, the probability that the sample mean sales price will be between $250,000 and $285,000 is 0.8893.

Option (d) is correct.

Please rate the answer. Thank you.


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