Question

Use 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

0.45
0.45

0.67
0.67

0.76
0.76

0.82

Answer 1

To calculate the standard deviation, we first need to determine the mean (average) of the values given and then use that to find the variance and standard deviation.

1. **Calculate the mean:**

Using the values and total observations:
- For 3 miles: 2 days
- For 4 miles: 3 days
- For 5 miles: 2 days

The total number of observations (days) is:
\[
\text{Total Days} = 2 + 3 + 2 = 7
\]

Now, calculate the weighted average (mean):
\[
\text{Mean} = \frac{(3 \times 2) + (4 \times 3) + (5 \times 2)}{7}
\]
\[
= \frac{(6) + (12) + (10)}{7} = \frac{28}{7} = 4
\]

2. **Calculate the variance:**

The variance is calculated using the formula:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N}
\]
where \(x_i\) are the values, \(\mu\) is the mean, \(f_i\) is the frequency (total observations), and \(N\) is the total observations.

Now calculate \((x_i - \mu)^2\) for each value:

- For 3 miles:
\[
(3 - 4)^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2
\]
- For 4 miles:
\[
(4 - 4)^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0
\]
- For 5 miles:
\[
(5 - 4)^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2
\]

Now sum these values:
\[
\text{Total} = 2 + 0 + 2 = 4
\]

Now calculate the variance:
\[
\sigma^2 = \frac{4}{7} \approx 0.57142857
\]

3. **Calculate the standard deviation:**
\[
\sigma = \sqrt{\sigma^2} = \sqrt{\frac{4}{7}} \approx 0.75
\]

Rounding this to the nearest hundredth gives approximately \(0.75\).

Based on the responses given:

- **0.45**
- **0.67**
- **0.76**
- **0.82**

The closest response to our calculated standard deviation \(0.75\) is **0.76**.