Question

the average miles per gallon of a particular automobile model are approximately normally distributed with a given mean μ = 43.8 miles per gallon and standard deviation σ = 5.1 miles per gallon. what percentage of the automobiles have an average miles per gallon between 38.7 miles per gallon and 48.9 miles per gallon?
○ 68%
○ 75%
○ 95%
○ 100%

Explanation:

Step1: Identify the number of standard deviations from the mean

First, we calculate how many standard deviations 38.7 and 48.9 are from the mean \(\mu = 43.8\) and standard deviation \(\sigma = 5.1\).

For 38.7:

$$ \frac{43.8 - 38.7}{5.1} = \frac{5.1}{5.1} = 1 $$

So, 38.7 is 1 standard deviation below the mean.

For 48.9:

$$ \frac{48.9 - 43.8}{5.1} = \frac{5.1}{5.1} = 1 $$

So, 48.9 is 1 standard deviation above the mean.

Step2: Apply the empirical rule for normal distribution

The empirical rule (also known as the 68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately 68% of the data falls within 1 standard deviation of the mean (\(\mu \pm \sigma\)).
• Approximately 95% of the data falls within 2 standard deviations of the mean (\(\mu \pm 2\sigma\)).
• Approximately 99.7% of the data falls within 3 standard deviations of the mean (\(\mu \pm 3\sigma\)).

Since 38.7 is \(\mu - \sigma\) and 48.9 is \(\mu + \sigma\), the percentage of data between them is approximately 68% according to the empirical rule.

Answer:

68%