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	2
1	15
2	30
3	27
4	10
5	8
6	8
Find the number that is 1.5 standard deviations BELOW the mean. (Round your answer to three decimal places.)
 


To find the value that is 1.5 standard deviations below the mean, we need to first calculate the mean (expected value) and the standard deviation of the given data. Here's the procedure:

### 1. Calculate the Mean (\(\mu\)):

The formula for the mean of a discrete probability distribution is:

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

where:
- \( X \) is the number of days per week,
- \( f \) is the frequency of each \( X \),
- \( n \) is the total number of observations (in this case, \( n = 100 \)).

#### Data:
\[
X = [0, 1, 2, 3, 4, 5, 6]
\]
\[
f = [2, 15, 30, 27, 10, 8, 8]
\]

Let's compute the mean.

### 2. Calculate the Standard Deviation (\(\sigma\)):

The formula for standard deviation is:

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

We'll compute this after calculating the mean.

### 3. Find 1.5 Standard Deviations Below the Mean:

To find the value 1.5 standard deviations below the mean, we calculate:

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

I'll calculate this step by step.### Results:
1. Mean (\(\mu\)) = 2.840
2. Standard Deviation (\(\sigma\)) = 1.481
3. The value that is 1.5 standard deviations below the mean is approximately **0.618**.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the value 2 standard deviations above the mean in this data set?
2. How do you calculate variance before finding standard deviation?
3. What percentage of clients use the facility 4 or more times a week?
4. What is the cumulative frequency of clients visiting less than 3 times a week?
5. How would the mean change if we removed the clients who never use the facility?

**Tip**: Always remember to use frequencies when calculating mean and standard deviation for grouped data.

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: 2, 1: 15, 2: 30, 3: 27, 4: 10, 5: 8, 6: 8. Find the number that is 1.5 standard deviations BELOW the mean. (Round your answer to three decimal places.)

Learn how to calculate the value that is 1.5 standard deviations below the mean for a dataset on exercise facility usage. This problem involves calculating the mean and standard deviation using frequency data for a sample of 100 clients. Suitable for high school students (Grades 9-12).

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