Respuesta :
Answer:
A. 0.9756
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 normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation(which is the square root of the variance) [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.
In this problem, we have that:
[tex]\mu = 2.77, \sigma = \sqrt{0.32} = 0.5657, n = 72, s = \frac{0.5657}{\sqrt{72}} = 0.0667[/tex]
What is the probability that a sample of 72 gas stations taken that same week will have a sample mean within $0.15 of the population mean?
This is the pvalue of Z when X = 2.77 + 0.15 = 2.92 subtracted by the pvalue of Z when X = 2.77 - 0.15 = 2.62. So
X = 2.92
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{2.92 - 2.77}{0.0667}[/tex]
[tex]Z = 2.25[/tex]
[tex]Z = 2.25[/tex] has a pvalue of 0.9878
X = 2.62
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{2.62 - 2.77}{0.0667}[/tex]
[tex]Z = -2.25[/tex]
[tex]Z = -2.25[/tex] has a pvalue of 0.0122
0.9878 - 0.0122 = 0.9756
