https://file.aiquickdraw.com/mathbot/file/1727447863609gy476k7p.jpg

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.

Car fuel mileage (mpg) is known to be normally distributed with a mean of 35 mpg and a standard deviation of 7 mpg. Based on this information, answer the following questions: 1. Approximately 95% of all cars have gas mileages between what two values? 2. Approximately what percentage of cars have gas mileages between 14 mpg and 56 mpg? 3. Approximately what percentage of cars have gas mileages above 35 mpg?

Solve problems related to the normal distribution of car fuel mileage with a mean of 35 mpg and a standard deviation of 7 mpg. Learn how to find the percentage of cars between certain mileages and understand the empirical rule (68-95-99.7 Rule). Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation