Fuel economy estimates for automobiles built in a certain year predicted a mean of 25.2 mpg and a standard deviation of 6.8 mpg for highway driving. Assume that a normal distribution can be applied. Within what range are 99.7% of the​ automobiles?

To find the range within which 99.7% of automobiles fall, we can use the empirical rule, which states that for a normal distribution:

- Approximately 68% of values fall within one standard deviation (μ ± σ).
- Approximately 95% of values fall within two standard deviations (μ ± 2σ).
- Approximately 99.7% of values fall within three standard deviations (μ ± 3σ).

Given:
- Mean (μ) = 25.2 mpg
- Standard deviation (σ) = 6.8 mpg

To find the range for 99.7%, we calculate:

1. Lower limit: \( \mu - 3\sigma = 25.2 - 3(6.8) \)
2. Upper limit: \( \mu + 3\sigma = 25.2 + 3(6.8) \)

Calculating these:

1. Lower limit: 
   \[
   25.2 - 20.4 = 4.8 \text{ mpg}
   \]

2. Upper limit:
   \[
   25.2 + 20.4 = 45.6 \text{ mpg}
   \]

Thus, the range within which 99.7% of the automobiles fall is **4.8 mpg to 45.6 mpg**.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How is the empirical rule applied in other statistical contexts?
2. What factors can affect the fuel economy of automobiles?
3. How can you calculate the percentage of automobiles that fall outside this range?
4. What is the significance of standard deviation in this context?
5. How would the range change if the standard deviation were larger? 

**Tip:** When working with normal distributions, always check if the data follows this pattern before applying these rules.

Fuel economy estimates for automobiles built in a certain year predicted a mean of 25.2 mpg and a standard deviation of 6.8 mpg for highway driving. Assume that a normal distribution can be applied. Within what range are 99.7% of the automobiles?

Explore the calculation of the fuel economy range for automobiles with a mean of 25.2 mpg and a standard deviation of 6.8 mpg, determining that 99.7% of vehicles fall between 4.8 mpg and 45.6 mpg. Learn about the empirical rule and its applications.

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