Let us assume that the prices of regular unleaded gasoline across the nation are
ID: 3150749 • Letter: L
Question
Let us assume that the prices of regular unleaded gasoline across the nation are normally distributed with a mean of $1.79 and a standard deviation of $0.25.
(a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all regular unleaded gasoline prices (hereafter referred to simply as gas prices). This would by symmetric, bell-shapped distribution with the peak centered above the horizontal scale value of 1.79 and standard deviated scale values marked at .25 unit intervals.
(b) If all possible samples of size 48 from the population of these gas prices are drawn and the mean is found for each sample, describe the shape and horizontal scaling on the graph of the sampling distribution for these sample mean values as theorized by the Central Limit Theorem.
(c) Find the probability that the price from a single randomly selected gas station will be more than $2.00. Based upon your results, would it be unusual to find an individual gas station where the price is more than $2.00? Explain.
(d) Find the probability that the mean from 15 randomly selected gas stations will be more than $2.00. Based upon your results, would it be unusual to find a sample of 15 randomly selected gas stations where the average price is more than $2.00? Explain.
Explanation / Answer
(a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all regular unleaded gasoline prices (hereafter referred to simply as gas prices).
This would by symmetric, bell-shapped distribution with the peak centered above the horizontal scale value of 1.79 and standard deviated scale values marked at .25 unit intervals.
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(b) If all possible samples of size 48 from the population of these gas prices are drawn and the mean is found for each sample, describe the shape and horizontal scaling on the graph of the sampling distribution for these sample mean values as theorized by the Central Limit Theorem.
Everything will be the same except the standard deviation is reduced,
sigma(X) = sigma/sqrt(n) = 0.25/sqrt(48) = 0.036084392
Hence,
This would by symmetric, bell-shapped distribution with the peak centered above the horizontal scale value of 1.79 and standard deviated scale values marked at 0.036084392 unit intervals. [CONCLUSION]
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(c) Find the probability that the price from a single randomly selected gas station will be more than $2.00. Based upon your results, would it be unusual to find an individual gas station where the price is more than $2.00? Explain.
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 2
u = mean = 1.79
s = standard deviation = 0.25
Thus,
z = (x - u) / s = 0.84
Thus, using a table/technology, the right tailed area of this is
P(z > 0.84 ) = 0.200454193 [ANSWER]
NO, as P > 0.05. [ANSWER]
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(d) Find the probability that the mean from 15 randomly selected gas stations will be more than $2.00. Based upon your results, would it be unusual to find a sample of 15 randomly selected gas stations where the average price is more than $2.00? Explain.
We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as
x = critical value = 2
u = mean = 1.79
n = sample size = 15
s = standard deviation = 0.25
Thus,
z = (x - u) * sqrt(n) / s = 3.253306011
Thus, using a table/technology, the right tailed area of this is
P(z > 3.253306011 ) = 0.000570353 [ANSWER]
Yes, as P < 0.05. [ANSWER]
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