Math Problem Statement
Suppose that mean retail price per gallon of regular grade gasoline is $3.42 with a standard deviation of $0.10 and that the retail price per gallon has a bell-shaped distribution. What percentage of regular grade gasoline sells for between $3.22 and $3.52 per gallon
Solution
To determine the percentage of regular grade gasoline that sells for between $3.22 and $3.52 per gallon, we can again use the Empirical Rule and z-scores.
Given:
- Mean (μ): $3.42
- Standard deviation (σ): $0.10
- Range: $3.22 to $3.52
Steps:
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Calculate the z-scores for the boundaries of $3.22 and $3.52.
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Interpret the z-scores:
- A z-score of -2 corresponds to 2 standard deviations below the mean.
- A z-score of 1 corresponds to 1 standard deviation above the mean.
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Apply the Empirical Rule:
- Approximately 68% of the data in a normal distribution falls within 1 standard deviation of the mean (from to ).
- Approximately 95% of the data falls within 2 standard deviations of the mean (from to ).
To find the percentage between and :
- From to covers 95% of the distribution.
- From to covers approximately 13.5% (half of the range from to ).
So, the percentage between and is:
- (from to ) + (from to )
Therefore, approximately 81.5% of regular grade gasoline sells for between $3.22 and $3.52 per gallon.
Would you like more details or have any questions?
Related Questions:
- What percentage of gasoline prices fall between $3.32 and $3.52?
- How much of the gasoline is expected to be priced below $3.22 per gallon?
- What is the percentage of gasoline prices between $3.42 and $3.62?
- How would the percentage change if the range was $3.32 to $3.62?
- What is the probability that a randomly selected gallon of gasoline costs more than $3.52?
Tip:
When using the Empirical Rule, breaking down the ranges into segments of standard deviations helps to determine the cumulative percentage accurately.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Z-score
Empirical Rule
Formulas
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Theorems
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Suitable Grade Level
Advanced High School / College
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