Respuesta :
Answer:
The minimum sizes of the homes the contractor should build is of 1692.24 square feet.
The maximum sizes of the homes the contractor should build is of 1927.76 square feet.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
The average size of homes built is 1810 ft2 with a standard deviation of 92 ft2
This means that [tex]\mu = 1810, \sigma = 92[/tex]
Middle 80%:
Between the (100-80)/2 = 10th percentile and the (100+80)/2 = 90th percentule.
Minimum size:
The 10th percentule, which is X when Z has a pvalue of 0.1, so X when Z = -1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.28 = \frac{X - 1810}{92}[/tex]
[tex]X - 1810 = -1.28*92[/tex]
[tex]X = 1692.24[/tex]
The minimum sizes of the homes the contractor should build is of 1692.24 square feet.
Maximum size:
The 90th percentule, which is X when Z has a pvalue of 0.9, so X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 1810}{92}[/tex]
[tex]X - 1810 = 1.28*92[/tex]
[tex]X = 1927.76[/tex]
The maximum sizes of the homes the contractor should build is of 1927.76 square feet.
