Question

For the following data set, calculate the percentage of data points that fall within one standard deviation of the mean, and compare the result to the expected percentage of a normal distribution. {50, 46, 54, 51, 29, 52, 48, 54, 47, 48} (1 point) Responses 80%; This percentage is lower than expected in a normal distribution. 80%; This percentage is lower than expected in a normal distribution. 80%; This percentage is higher than expected in a normal distribution. 80%; This percentage is higher than expected in a normal distribution. 90%; This percentage is lower than expected in a normal distribution. 90%; This percentage is lower than expected in a normal distribution. 90%; This percentage is higher than expected in a normal distribution. 90%; This percentage is higher than expected in a normal distribution.

Answer 1

To calculate the percentage of data points that fall within one standard deviation of the mean for the given data set {50, 46, 54, 51, 29, 52, 48, 54, 47, 48}, follow these steps:

1. **Calculate the mean:**
\[
\text{Mean} = \frac{50 + 46 + 54 + 51 + 29 + 52 + 48 + 54 + 47 + 48}{10} = \frac{481}{10} = 48.1
\]

2. **Calculate the standard deviation:**
- First, find the deviations from the mean for each data point, square them, and calculate the variance:
\[
\text{Deviations} = (50 - 48.1)^2, (46 - 48.1)^2, (54 - 48.1)^2, (51 - 48.1)^2, (29 - 48.1)^2,
(52 - 48.1)^2, (48 - 48.1)^2, (54 - 48.1)^2, (47 - 48.1)^2, (48 - 48.1)^2
\]
\[
= (1.9)^2, (-2.1)^2, (5.9)^2, (2.9)^2, (-19.1)^2, (3.9)^2, (-0.1)^2, (5.9)^2, (-1.1)^2, (-0.1)^2
\]
\[
= 3.61, 4.41, 34.81, 8.41, 364.81, 15.21, 0.01, 34.81, 1.21, 0.01
\]

\[
\text{Variance} = \frac{3.61 + 4.41 + 34.81 + 8.41 + 364.81 + 15.21 + 0.01 + 34.81 + 1.21 + 0.01}{10} = \frac{473.3}{10} = 47.33
\]
\[
\text{Standard Deviation} = \sqrt{47.33} \approx 6.88
\]

3. **Find the range within one standard deviation of the mean:**
- Mean ± Standard Deviation:
\[
\text{Lower Bound} = 48.1 - 6.88 \approx 41.22
\]
\[
\text{Upper Bound} = 48.1 + 6.88 \approx 54.98
\]

4. **Count the data points within that range:**
- Data points: 50, 46, 54, 51, 29, 52, 48, 54, 47, 48
- Points within [41.22, 54.98]: {50, 46, 54, 51, 52, 48, 54, 47, 48} (9 data points)

5. **Calculate the percentage:**
\[
\text{Percentage} = \left(\frac{9}{10}\right) \times 100 = 90\%
\]

The expected percentage of data points falling within one standard deviation of the mean in a normal distribution is approximately 68%.

**Result:**
90%; This percentage is higher than expected in a normal distribution.