Let's go through each of the questions and solve them using the information provided:

Given Information:

• The fuel mileage (mpg) is normally distributed.
• Mean ($\mu$) = 35 mpg.
• Standard deviation ($\sigma$) = 7 mpg.

Question 8:

Approximately 95% of all cars have gas mileages between what two values?

For a normal distribution, 95% of the data lies within 2 standard deviations of the mean. This means we need to calculate the range for $\mu \pm 2\sigma$.

$\text{Lower bound} = \mu - 2\sigma = 35 - 2(7) = 35 - 14 = 21$ $\text{Upper bound} = \mu + 2\sigma = 35 + 2(7) = 35 + 14 = 49$

Thus, 95% of cars have mileages between 21 and 49 mpg.

The correct answer is (a) 21 and 49.

Question 9:

Approximately what percentage of cars have gas mileages between 14 mpg and 56 mpg?

We need to check how many standard deviations 14 mpg and 56 mpg are from the mean:

• For 14 mpg: $Z = \frac{14 - 35}{7} = \frac{-21}{7} = -3$ So, 14 mpg is 3 standard deviations below the mean.

• For 56 mpg: $Z = \frac{56 - 35}{7} = \frac{21}{7} = 3$ So, 56 mpg is 3 standard deviations above the mean.

From the empirical rule, about 99.7% of data lies within 3 standard deviations of the mean in a normal distribution.

The correct answer is (a) 99.7%.

Question 10:

Approximately what percentage of cars have gas mileages above 35 mpg?

Since the mean is 35 mpg, and normal distribution is symmetric around the mean, 50% of the data lies above the mean.

The correct answer is (a) 50%.

Would you like further details on any of these answers or concepts?

Here are 5 related questions to deepen understanding:

1. What percentage of cars have mileages below 21 mpg?
2. How do we use Z-scores to find percentages in a normal distribution?
3. What percentage of cars have gas mileages between 28 mpg and 42 mpg?
4. If the standard deviation were smaller, how would the ranges in questions 8 and 9 change?
5. How does changing the mean affect the normal distribution curve?

Tip: The empirical rule (68-95-99.7 rule) is a powerful shortcut for understanding how data is distributed in a normal distribution.