Respuesta :
Answer:
P(939.6 < X < 972.5) = 0.6469
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
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 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 normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
CNNBC recently reported that the mean annual cost of auto insurance is 965 dollars. Assume the standard deviation is 113 dollars.
This means that [tex]\mu = 965, \sigma = 113[/tex]
Sample of 57:
This means that [tex]n = 57, s = \frac{113}{\sqrt{57}} = 14.97[/tex]
Find the probability that a single randomly selected policy has a mean value between 939.6 and 972.5 dollars.
This is the pvalue of Z when X = 972.5 subtracted by the pvalue of Z when X = 939.6. So
X = 972.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{972.5 - 965}{14.97}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a pvalue of 0.6915
X = 939.6
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{939.6 - 965}{14.97}[/tex]
[tex]Z = -1.7[/tex]
[tex]Z = -1.7[/tex] has a pvalue of 0.0446
0.6915 - 0.0446 = 0.6469
So
P(939.6 < X < 972.5) = 0.6469
