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}
(1 point)
Responses
40%; This percentage is lower than expected in a normal distribution.
40%; This percentage is lower than expected in a normal distribution.
50%; This percentage is lower than expected in a normal distribution.
50%; This percentage is lower than expected in a normal distribution.
60%; This percentage is lower than expected in a normal distribution.
60%; This percentage is lower than expected in a normal distribution.
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 find the mean and standard deviation of the data set.
Mean:
The mean is calculated by summing up all the values and dividing by the total number of values.
Mean = (8 + 12 + 27 + 32 + 45 + 57 + 61 + 73 + 82 + 94) / 10 = 481 / 10 = 48.1.
Standard Deviation:
The standard deviation is a measure of how spread out the data points are from the mean. It is calculated using the formula:
Standard Deviation = sqrt((sum((x - mean)^2)) / (n - 1)),
where x is each value in the data set, mean is the calculated mean, and n is the total number of values.
Standard Deviation = sqrt((sum((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 - 1))
= sqrt(9973.9 / 9) = sqrt(1108.21) = 33.27.
Now, we can determine which values fall within one standard deviation of the mean:
Lower Bound = Mean - Standard Deviation = 48.1 - 33.27 = 14.83
Upper Bound = Mean + Standard Deviation = 48.1 + 33.27 = 81.37
The values 27, 32, 45, 57, 61, 73, and 82 fall within the range of 14.83 to 81.37. There are 7 values that fall within this range.
Percentage = (Number of values within one standard deviation / Total number of values) * 100 = (7 / 10) * 100 = 70%.
Therefore, the correct answer is:
70%; This percentage is close to the expected percentage in a normal distribution.