Understanding Auto Fuel Economy Using the 68-95-99.7 Rule

Math Problem Statement

Fuel economy estimates for automobiles built one year predicted a mean of

26.226.2

mpg and a standard deviation of

5.85.8

mpg for highway driving. Assume that a Normal model can be applied. Use the

68minus−95minus−99.7

Rule to complete parts a) through e).

Question content area bottom

Part 1

a right parenthesisa)

Draw the model for auto fuel economy.

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A bell-shaped curve with horizontal axis from less than 8.8 to 61 plus begins just above the horizontal axis, increases at an increasing and then decreasing rate to its maximum at 26.2, and decreases at an increasing and then decreasing rate approaching the horizontal axis. The seven equidistant horizontal axis labels are as follows from left to right: 8.8, 14.6, 20.4, 26.2, 37.8, 49.4, 61. The curve is visually symmetric. The area below the curve is subdivided into regions by the intervals and labeled as follows: from 20.4 to 37.8, "68%"; from 14.6 to 49.4, "95%"; from 8.8 to 61, "99.7%."

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A bell-shaped curve with horizontal axis from less than 8.8 to 43.6 plus in intervals of 5.8 begins just above the horizontal axis, increases at an increasing and then decreasing rate to its maximum at 26.2, and decreases at an increasing and then decreasing rate approaching the horizontal axis. The curve is symmetric. The area below the curve is subdivided into regions by the intervals and labeled as follows: from 20.4 to 32, "68%"; from 14.6 to 37.8, "95%"; from 8.8 to 43.6, "99.7%."

Your answer is correct.

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A bell-shaped curve with horizontal axis from less than 17.5 to 34.9 plus in intervals of 2.9 begins just above the horizontal axis, increases at an increasing and then decreasing rate to its maximum at 26.2, and decreases at an increasing and then decreasing rate approaching the horizontal axis. The curve is symmetric. The area below the curve is subdivided into regions by the intervals and labeled as follows: from 23.3 to 29.1, "68%"; from 20.4 to 32, "95%"; from 17.5 to 34.9, "99.7%."

Part 2

In what interval would you expect the central

9595%

of autos to be found?

Using the 68-95-99.7 rule, the central

9595%

of autos can be expected to be found in the interval from

14.614.6

37.837.8

mpg.

(Do not round. Type integers or decimals.)

Part 3

About what percent of autos should get more than

3232

mpg?

Using the 68-95-99.7 rule, about

enter your response here%

of autos should get more than

3232

mpg.

(Do not round. Type an integer or a decimal.)

Solution

To tackle this problem, we'll use the 68-95-99.7 rule (also known as the empirical rule), which applies to data that is normally distributed.

Part 1: Drawing the model for auto fuel economy

Given:

Mean ( $\mu$ ) = 26.2 mpg
Standard deviation ( $\sigma$ ) = 5.8 mpg

According to the 68-95-99.7 rule:

68% of the data falls within 1 standard deviation of the mean (i.e., between $\mu - \sigma$ and $\mu + \sigma$ ).
95% of the data falls within 2 standard deviations of the mean (i.e., between $\mu - 2\sigma$ and $\mu + 2\sigma$ ).
99.7% of the data falls within 3 standard deviations of the mean (i.e., between $\mu - 3\sigma$ and $\mu + 3\sigma$ ).

Let's calculate these intervals:

1 standard deviation: $\mu \pm \sigma = 26.2 \pm 5.8$ → [20.4, 32.0] mpg
2 standard deviations: $\mu \pm 2\sigma = 26.2 \pm 2(5.8)$ → [14.6, 37.8] mpg
3 standard deviations: $\mu \pm 3\sigma = 26.2 \pm 3(5.8)$ → [8.8, 43.6] mpg

Correct answer: Based on the choices, the correct model corresponds to Option B. The model includes labels for 20.4, 14.6, 8.8, 37.8, 32, 43.6, and 26.2 along the horizontal axis, with corresponding areas of 68%, 95%, and 99.7%.

Part 2: Interval for the central 95% of autos

For the central 95%:

This corresponds to the interval within 2 standard deviations of the mean: [14.6, 37.8] mpg.

Answer: The central 95% of autos can be expected to be found in the interval from 14.6 to 37.8 mpg.

Part 3: Percentage of autos getting more than 32 mpg

For autos getting more than 32 mpg:

32 mpg is 1 standard deviation above the mean (i.e., $\mu + \sigma$ ).
Using the 68-95-99.7 rule, 68% of autos fall within 1 standard deviation (i.e., between 20.4 and 32.0 mpg), so 34% fall above the mean and 34% fall below the mean.
Therefore, for autos getting more than 32 mpg, this percentage corresponds to half of the remaining 32% of autos beyond 1 standard deviation. That is, $\frac{32\%}{2} = 16\%$ .

Answer: About 16% of autos should get more than 32 mpg.

Would you like more details or have any other questions?

Here are 5 related questions to further your understanding:

What percentage of autos would you expect to get less than 20.4 mpg?
How would the intervals change if the standard deviation were larger?
Can you calculate the z-score for an auto that gets 40 mpg?
What does the 68-95-99.7 rule tell us about the distribution of extreme values?
How does the empirical rule relate to the concept of a normal distribution?

Tip: Understanding the 68-95-99.7 rule is crucial for working with normally distributed data, as it allows you to quickly estimate the proportion of values within certain ranges.

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Math Problem Analysis

Mathematical Concepts

Normal Distribution
Statistics

Formulas

68-95-99.7 Rule

Theorems

Suitable Grade Level

High School

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