Question

Use the table to answer the question.

Minutes the Bus Came Late −1
−
1
0 1 2 3
Number of Days 1 2 4 2 1
Addie’s bus pickup time is 8:00. The table shows the minutes before and after 8:00 the bus arrived at the stop for the last 10 days. Negative values represent minutes before 8:00. Calculate the standard deviation for the number of minutes the bus arrived before or after 8:00. Round your answer to the nearest tenth.

(1 point)
The standard deviation is 1.1.

Answer 1

To calculate the standard deviation, we need to follow these steps:

1. **List the data points**: Based on the table, we can create the data points based on the "Number of Days":
- For -1 minutes: 1 day
- For 0 minutes: 2 days
- For 1 minute: 4 days
- For 2 minutes: 2 days
- For 3 minutes: 1 day

Thus, the data set is:
\(-1, 0, 0, 1, 1, 1, 1, 2, 2, 3\).

2. **Calculate the mean**:
\[
\text{Mean} (\mu) = \frac{\text{Sum of all data points}}{\text{Total number of data points}}
\]
\[
\text{Mean} = \frac{(-1) + (0 \cdot 2) + (1 \cdot 4) + (2 \cdot 2) + (3)}{10} = \frac{-1 + 0 + 4 + 4 + 3}{10} = \frac{10}{10} = 1
\]

3. **Calculate the variance**:
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) are the data points and \(N\) is the total number of data points.

Calculate each variance component:
- For \(-1\): \((-1 - 1)^2 = (-2)^2 = 4\)
- For \(0\): \((0 - 1)^2 = (-1)^2 = 1\) (2 days, so multiply by 2)
- For \(1\): \((1 - 1)^2 = (0)^2 = 0\) (4 days, so multiply by 4)
- For \(2\): \((2 - 1)^2 = (1)^2 = 1\) (2 days, so multiply by 2)
- For \(3\): \((3 - 1)^2 = (2)^2 = 4\)

Now calculate the total variance:
\[
\text{Total Sum} = 4 + (1 \cdot 2) + (0 \cdot 4) + (1 \cdot 2) + 4 = 4 + 2 + 0 + 2 + 4 = 12
\]
\[
\text{Variance} = \frac{12}{10} = 1.2
\]

4. **Calculate the standard deviation**:
\[
\text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} = \sqrt{1.2} \approx 1.095
\]

Rounding to the nearest tenth:
The standard deviation is approximately **1.1**.

So your original answer is confirmed: the standard deviation is **1.1**.