Respuesta :
Answer:
90.66% probability that over the next 65 houses the realtor sells, the mean value is at least $225,000.
Step-by-step explanation:
To solve this problem, it is important to kknow the Normal probability distribution and the Central Limit Theorem.
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 234000, \sigma = 55000, n = 65, s = \frac{55000}{\sqrt{65}} = 6821.91[/tex]
What is the probability that over the next 65 houses the realtor sells, the mean value is at least $225,000.
This probability is 1 subtracted by the pvalue of Z when X = 225. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem, we have that:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{225000 - 234000}{6821.91}[/tex]
[tex]Z = -1.32[/tex]
[tex]Z = -1.32[/tex] has a pvalue of 0.0934.
So there is a 1-0.0934 = 0.9066 = 90.66% probability that over the next 65 houses the realtor sells, the mean value is at least $225,000.
