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:

  1. Calculate the z-scores for the boundaries of $3.22 and $3.52.

    z1=3.223.420.10=0.200.10=2z_1 = \frac{3.22 - 3.42}{0.10} = \frac{-0.20}{0.10} = -2 z2=3.523.420.10=0.100.10=1z_2 = \frac{3.52 - 3.42}{0.10} = \frac{0.10}{0.10} = 1
  2. 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.
  3. Apply the Empirical Rule:

    • Approximately 68% of the data in a normal distribution falls within 1 standard deviation of the mean (from z=1z = -1 to z=1z = 1).
    • Approximately 95% of the data falls within 2 standard deviations of the mean (from z=2z = -2 to z=2z = 2).

To find the percentage between z=2z = -2 and z=1z = 1:

  • From z=2z = -2 to z=2z = 2 covers 95% of the distribution.
  • From z=2z = -2 to z=1z = -1 covers approximately 13.5% (half of the range from z=2z = -2 to z=1z = -1).

So, the percentage between z=2z = -2 and z=1z = 1 is:

  • 68%68\% (from z=1z = -1 to z=1z = 1) + 13.5%13.5\% (from z=2z = -2 to z=1z = -1)

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:

  1. What percentage of gasoline prices fall between $3.32 and $3.52?
  2. How much of the gasoline is expected to be priced below $3.22 per gallon?
  3. What is the percentage of gasoline prices between $3.42 and $3.62?
  4. How would the percentage change if the range was $3.32 to $3.62?
  5. 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