Question

se the table to answer the question.

Value 3 4 5
Total Observations 2 3 2
Charles records the number of miles he walks every day. The chart includes how many days he walked the given number of miles for one week. Calculate the standard deviation for the number of miles he walked. Round the answer to the nearest hundredth.

(1 point)
Responses

0.76
0.76

0.45
0.45

0.82
0.82

0.67

Answer 1

To calculate the standard deviation for the number of miles Charles walked based on the provided data, we first need to find the mean, variance, and then take the square root of the variance.

### Data Summary

- For 3 miles: 2 days
- For 4 miles: 3 days
- For 5 miles: 2 days

### Step 1: Calculate the Mean

The formula for the mean (μ) is:

\[
\mu = \frac{\sum (x \cdot f)}{N}
\]

where:
- \( x \) is the value (miles walked)
- \( f \) is the frequency (number of days)
- \( N \) is the total number of observations

Calculating total observations (N):

\[
N = 2 + 3 + 2 = 7 \text{ days}
\]

Calculating \( \sum (x \cdot f) \):

\[
\sum (x \cdot f) = (3 \cdot 2) + (4 \cdot 3) + (5 \cdot 2) = 6 + 12 + 10 = 28
\]

Now calculate the mean:

\[
\mu = \frac{28}{7} = 4
\]

### Step 2: Calculate the Variance

The formula for variance (\( \sigma^2 \)) is:

\[
\sigma^2 = \frac{\sum f \cdot (x - \mu)^2}{N}
\]

Calculating \( (x - \mu)^2 \) for each value:

- For 3 miles:
\[
(3 - 4)^2 = (-1)^2 = 1
\]
Contribution to the variance:
\[
2 \cdot 1 = 2
\]

- For 4 miles:
\[
(4 - 4)^2 = (0)^2 = 0
\]
Contribution to the variance:
\[
3 \cdot 0 = 0
\]

- For 5 miles:
\[
(5 - 4)^2 = (1)^2 = 1
\]
Contribution to the variance:
\[
2 \cdot 1 = 2
\]

Now, summing these contributions:

\[
\sum f \cdot (x - \mu)^2 = 2 + 0 + 2 = 4
\]

Finally, calculate the variance:

\[
\sigma^2 = \frac{4}{7} \approx 0.57
\]

### Step 3: Calculate the Standard Deviation

The standard deviation (\( \sigma \)) is the square root of the variance:

\[
\sigma = \sqrt{0.57} \approx 0.76
\]

### Final Answer

The standard deviation for the number of miles he walked, rounded to the nearest hundredth, is:

**0.76**

So, the correct response is:

**0.76**