The results of a certain medical test are normally distributed with a mean of 120 and a standard deviation of 6. Convert the given results into​ z-scores, and then use the accompanying table of​ z-scores and percentiles to find the percentage of people with readings between 115 and 125. 
LOADING... Click the icon to view the table of​ z-scores and percentiles.
Question content area bottom
Part 1
The percentage of people with readings between 115 and 125 is 

Let's solve this problem step-by-step:

### Step 1: Calculate the z-scores
To convert the values into z-scores, we'll use the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is the data point
- \( \mu \) is the mean
- \( \sigma \) is the standard deviation

Given:
- Mean, \( \mu = 120 \)
- Standard deviation, \( \sigma = 6 \)

Let's find the z-scores for 115 and 125.

#### For \( X = 115 \):
\[
z = \frac{115 - 120}{6} = \frac{-5}{6} \approx -0.83
\]

#### For \( X = 125 \):
\[
z = \frac{125 - 120}{6} = \frac{5}{6} \approx 0.83
\]

### Step 2: Use the z-score table to find the percentage of values between these z-scores
Using the z-score table:
- The z-score of \( -0.83 \) corresponds to a percentile of approximately 20.34%.
- The z-score of \( 0.83 \) corresponds to a percentile of approximately 79.66%.

To find the percentage of people with readings between 115 and 125, subtract the percentile for \( -0.83 \) from that of \( 0.83 \):
\[
79.66\% - 20.34\% = 59.32\%
\]

### Conclusion
The percentage of people with readings between 115 and 125 is approximately **59.32%**.

Would you like more details or have any questions? 

Here are some related questions that might help deepen your understanding:
1. What is a z-score and why is it useful in statistics?
2. How do you interpret a z-score in terms of standard deviations from the mean?
3. How does the area under the normal distribution curve relate to percentiles?
4. What are some examples of other real-life scenarios where z-scores might be applied?
5. How do you use a z-score table to find probabilities for values in a normal distribution?

**Tip:** Understanding z-scores helps in standardizing different datasets, allowing for comparisons across different scales or units.

The results of a certain medical test are normally distributed with a mean of 120 and a standard deviation of 6. Convert the given results into​ z-scores, and then use the accompanying table of​ z-scores and percentiles to find the percentage of people with readings between 115 and 125.

Learn how to calculate z-scores for a normal distribution with a mean of 120 and standard deviation of 6. This detailed guide explains how to convert raw values into z-scores and use a z-score table to find the percentage of data between 115 and 125. Perfect for high school statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation