Question

The energy information administration reported that the mean retail price per gallon of regular grade gasoline was $3.43. Suppose that standard deviation was $0.10 and that the retail price per gallon has a bell shaped distribution.
What percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon?
What percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon?
What percentage of regular grade gasoline sold for more than $3.63 per gallon?

Answer 1

Answer:

68.26% of regular grade gasoline sold between $3.33 and $3.53 per gallon

81.85% of regular grade gasoline sold between $3.33 and $3.63 per gallon

2.28% of regular grade gasoline sold for more than $3.63 per gallon

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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.

In this problem, we have that:

$\mu = 3.43, \sigma = 0.1$

What percentage of regular grade gasoline sold between $3.33 and $3.53 per gallon?

This is the pvalue of Z when X = 3.53 subtracted by the pvalue of Z when X = 3.33. So

X = 3.53

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.53 - 3.43}{0.1}$

$Z = 1$

$Z = 1$ has a pvalue of 0.8413

X = 3.33

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.33 - 3.43}{0.1}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587

0.8413 - 0.1587 = 0.6826

68.26% of regular grade gasoline sold between $3.33 and $3.53 per gallon

What percentage of regular grade gasoline sold between $3.33 and $3.63 per gallon?

This is the pvalue of Z when X = 3.53 subtracted by the pvalue of Z when X = 3.33. So

X = 3.63

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.63 - 3.43}{0.1}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

X = 3.33

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.33 - 3.43}{0.1}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587

0.9772 - 0.1587 = 0.8185

81.85% of regular grade gasoline sold between $3.33 and $3.63 per gallon

What percentage of regular grade gasoline sold for more than $3.63 per gallon?

This is 1 subtracted by the pvalue of Z when X = 3.63. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.63 - 3.43}{0.1}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

1 - 0.9772 = 0.0228

2.28% of regular grade gasoline sold for more than $3.63 per gallon