A sample of 100 clients of an exercise facility was selected. Let X = the number of days per week that a randomly selected client uses the exercise facility.
X	Frequency
0	3
1	15
2	32
3	27
4	11
5	7
6	5
Find the number that is 1.5 standard deviations BELOW the mean. (Round your answer to three decimal places.)
 
Incorrect: Your answer is incorrect.


To find the number that is 1.5 standard deviations below the mean for this data set, we need to follow these steps:

### Step 1: Compute the Mean
The mean \(\mu\) of a distribution is calculated as:

\[
\mu = \frac{\sum (X \times f)}{n}
\]

Where:
- \(X\) is the number of days per week,
- \(f\) is the frequency,
- \(n = 100\) is the total number of clients.

Using the given dataTo find the number that is 1.5 standard deviations below the mean for this data set, we need to follow these steps:

### Step 1: Compute the Mean
The mean \(\mu\) of a distribution is calculated as:

\[
\mu = \frac{\sum (X \times f)}{n}
\]

Where:
- \(X\) is the number of days per week,
- \(f\) is the frequency,
- \(n = 100\) is the total number of clients.

Using the given 
\[
\sum (X \times f) = (0 \times 3) + (1 \times 15) + (2 \times 32) + (3 \times 27) + (4 \times 11) + (5 \times 7) + (6 \times 5)
\]

Let's calculate the sum of the products and find the mean.

### Step 2: Compute the Variance and Standard Deviation
The variance is calculated as:

\[
\sigma^2 = \frac{\sum f(X - \mu)^2}{n}
\]

And the standard deviation is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

### Step 3: Compute 1.5 Standard Deviations Below the Mean
Once we have the mean and standard deviation, 1.5 standard deviations below the mean is:

\[
\text{Result} = \mu - 1.5 \times \sigma
\]

Let's calculate this now.The mean number of days per week that a randomly selected client uses the exercise facility is approximately 2.690. The standard deviation is approximately 1.391. Therefore, 1.5 standard deviations below the mean is approximately:

\[
0.604
\]

Would you like more details on any of the steps?

Here are 5 related questions to explore:
1. How is the variance different from the standard deviation in statistics?
2. What is the significance of standard deviations in a normal distribution?
3. How would the result change if the number of clients were larger?
4. Can we assume this distribution is normal based on the given data?
5. What does it mean for a value to be 1.5 standard deviations below the mean?

**Tip:** Standard deviations can give you insight into the spread or dispersion of data around the mean in a distribution.

A sample of 100 clients of an exercise facility was selected. Let X = the number of days per week that a randomly selected client uses the exercise facility. Find the number that is 1.5 standard deviations BELOW the mean. (Round your answer to three decimal places.) X: {0: 3, 1: 15, 2: 32, 3: 27, 4: 11, 5: 7, 6: 5}

Learn how to find 1.5 standard deviations below the mean for a sample of 100 clients at an exercise facility. Using statistics concepts like mean, variance, and standard deviation, this example involves calculating the mean number of days clients use the facility and applying standard deviation rules. Suitable for high school students in Grades 9-12.

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