An automobile manufacturer introduces a new model that averages 27.0 miles per gallon in the city. A person who plans to purchase one of these new cars wrote the manufacturer for the details of the tests, and found out that the standard deviation is 3.8 miles per gallon. Assume that in-city mileage is approximately normally distributed.
% of the automobiles have between 18 and 21 miles per gallon.
Round the z-score to two decimal places if needed. Round your answer to two decimal places. Do not include the % sign in your answer.

To determine the percentage of automobiles that have between 18 and 21 miles per gallon, we will follow these steps:

### Step 1: Convert 18 and 21 miles per gallon to z-scores
Given:
- Mean \( \mu = 27.0 \) mpg
- Standard deviation \( \sigma = 3.8 \) mpg

The z-score formula is:
\[
z = \frac{X - \mu}{\sigma}
\]
where \( X \) is the value we are converting.

#### For \( X = 18 \):
\[
z = \frac{18 - 27.0}{3.8} = \frac{-9.0}{3.8} \approx -2.37
\]

#### For \( X = 21 \):
\[
z = \frac{21 - 27.0}{3.8} = \frac{-6.0}{3.8} \approx -1.58
\]

### Step 2: Find the Probability for Each Z-Score
Using a z-table or a standard normal distribution calculator, we can find the probabilities for \( z = -2.37 \) and \( z = -1.58 \):

- The probability for \( z = -2.37 \) is approximately 0.0089.
- The probability for \( z = -1.58 \) is approximately 0.0571.

### Step 3: Calculate the Percentage between the Two Z-Scores
To find the percentage of automobiles with in-city mileage between 18 and 21 miles per gallon, we subtract these probabilities:
\[
0.0571 - 0.0089 = 0.0482
\]

### Step 4: Convert to a Percentage
\[
0.0482 \times 100 \approx 4.82
\]

### Final Answer
The percentage of automobiles that have between 18 and 21 miles per gallon is **4.82**.

Would you like further details or have any questions?

---

**Further Exploration Questions:**
1. What is the probability that a car has more than 30 miles per gallon?
2. What percentage of cars have less than 24 miles per gallon?
3. How would the answer change if the mean was 30 mpg with the same standard deviation?
4. What percentage of cars have between 25 and 29 miles per gallon?
5. How would the standard deviation affect the probability range?

**Tip:** When working with normal distributions, sketching the bell curve can help visualize the z-scores and probability areas.

An automobile manufacturer introduces a new model that averages 27.0 miles per gallon in the city. A person who plans to purchase one of these new cars wrote the manufacturer for the details of the tests, and found out that the standard deviation is 3.8 miles per gallon. Assume that in-city mileage is approximately normally distributed. % of the automobiles have between 18 and 21 miles per gallon. Round the z-score to two decimal places if needed. Round your answer to two decimal places. Do not include the % sign in your answer.

Learn to calculate the probability that an automobile achieves between 18 and 21 miles per gallon, given a mean of 27.0 mpg and a standard deviation of 3.8 mpg. Follow this step-by-step guide using the normal distribution and z-score formula. Perfect for 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