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.
{8, 12, 27, 32, 45, 57, 61, 73, 82, 94}
a. 40%; This percentage is lower than expected in a normal distribution.
b. 50%; This percentage is lower than expected in a normal distribution.
c. 60%; This percentage is lower than expected in a normal distribution.
d.70%; This percentage is close to the expected percentage in a normal distribution.
To calculate the percentage of data points that fall within one standard deviation of the mean, we first need to calculate the mean and the standard deviation for the given data set.
The mean (µ) for the data set is calculated by summing all the values and dividing by the number of data points:
µ = (8 + 12 + 27 + 32 + 45 + 57 + 61 + 73 + 82 + 94) / 10
µ = 481 / 10
µ = 48.1
To calculate the standard deviation (σ) for the data set, we first need to calculate the variance (σ^2), which is the average of the squared deviations from the mean.
The variance is calculated by subtracting the mean from each data point, squaring the result, summing all the squared values, and dividing by the number of data points:
Variance = [(8 - 48.1)^2 + (12 - 48.1)^2 + (27 - 48.1)^2 + (32 - 48.1)^2 + (45 - 48.1)^2 + (57 - 48.1)^2 + (61 - 48.1)^2 + (73 - 48.1)^2 + (82 - 48.1)^2 + (94 - 48.1)^2] / 10
Variance = [1607.29 + 1433.61 + 435.61 + 257.049 + 5.41 + 65.61 + 156.81 + 600.09 + 1290.49 + 1967.69] / 10
Variance = 9590.95 / 10
Variance = 959.095
The standard deviation is the square root of the variance:
Standard Deviation = √959.095
Standard Deviation ≈ 30.98
Now, we can calculate the percentage of data points that fall within one standard deviation of the mean. In a normal distribution, approximately 68% of data points fall within one standard deviation of the mean.
For the given data set, we can count the number of data points that fall within one standard deviation of the mean by checking if the data point is within the range of (mean - standard deviation) to (mean + standard deviation):
Data points within range: 27, 32, 45, 57, 61, 73, 82
Number of data points within range = 7
Percentage of data points within one standard deviation = (Number of data points within range / Total number of data points) * 100
Percentage of data points within one standard deviation = (7 / 10) * 100
Percentage of data points within one standard deviation = 70%
Comparing this result to the expected percentage of a normal distribution (68%), we can conclude that:
d. 70%; This percentage is close to the expected percentage in a normal distribution.
could it be c
No, the correct answer is d. 70%; This percentage is close to the expected percentage in a normal distribution.
The expected percentage of data points within one standard deviation of the mean in a normal distribution is approximately 68%. In this data set, 7 out of the 10 data points fall within one standard deviation of the mean, which corresponds to 70%. This is close to the expected percentage in a normal distribution.