In: Statistics and Probability
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
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.
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