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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 [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.
In this problem, we have that:
[tex]\mu = 3.43, \sigma = 0.1[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.53 - 3.43}{0.1}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a pvalue of 0.8413
X = 3.33
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.33 - 3.43}{0.1}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] 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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.63 - 3.43}{0.1}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
X = 3.33
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.33 - 3.43}{0.1}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] 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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{3.63 - 3.43}{0.1}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] 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
The retail price per gallon that has a bell shaped distribution is an illustration of a normal distribution
The percentage between $3.33 and $3.53
The given parameters are:
[tex]\bar x = 3.43[/tex]
[tex]\sigma = 0.10[/tex]
Start by calculating the z-scores using:
[tex]z = \frac{x - \bar x}{\sigma}[/tex]
When x = 3.33, we have:
[tex]z = \frac{3.33 - 3.43}{0.10} = -1[/tex]
When x = 3.53, we have:
[tex]z = \frac{3.53 - 3.43}{0.10} = 1[/tex]
The p-value at these z-scores are:
p = (z = -1) = 0.1587
p = (z = 1) = 0.8413
The percentage is then calculated as:
[tex]p = 0.8413 - 0.1587[/tex]
[tex]p = 0.6826[/tex]
[tex]p = 68.26\%[/tex]
68.26% of regular grade gasoline are sold between $3.33 and $3.53 per gallon
The percentage between $3.33 and $3.63
When x = 3.33, we have:
[tex]z = \frac{3.33 - 3.43}{0.10} = -1[/tex]
When x = 3.63, we have:
[tex]z = \frac{3.63 - 3.43}{0.10} = 2[/tex]
The p-value at these z-scores are:
p = (z = -1) = 0.1587
p = (z = 2) = 0.9772
The percentage is then calculated as:
[tex]p = 0.9772 - 0.1587[/tex]
[tex]p = 0.8185[/tex]
[tex]p = 81.85\%[/tex]
81.85% of regular grade gasoline are sold between $3.33 and $3.63 per gallon
The percentage for more than $3.63
When x = 3.63, we have:
[tex]z = \frac{3.63 - 3.43}{0.10} = 2[/tex]
The p-value at this z-score are:
p = (z = 2) = 0.9772
The percentage is then calculated as:
[tex]p =1- 0.9772[/tex]
[tex]p = 0.0228[/tex]
[tex]p = 2.28\%[/tex]
2.28% of regular grade gasoline are sold for more than $3.63 per gallon
Read more about z-scores at:
https://brainly.com/question/25638875
