Susan keeps track of the number of tickets sold for each play presented at the community theater. Within how many standard deviations of the mean do all the values fall? 135, 71, 69, 80, 158, 152, 161, 96, 122, 118, 87, 85 (1 point) Responses 3 3 4 4 5 5 2
The mean of the ticket sales is found by adding up all the values and dividing by the total number of values:
(135 + 71 + 69 + 80 + 158 + 152 + 161 + 96 + 122 + 118 + 87 + 85) / 12 = 122
Next, we need to find the standard deviation. The formula for standard deviation is:
sd = sqrt[(Σ(xi - x̄)^2) / n]
Where Σ(xi - x̄)^2 is the sum of the squared differences between each value and the mean, n is the number of values, and x̄ is the mean.
Calculating the standard deviation:
sd = sqrt[((135-122)^2 + (71-122)^2 + (69-122)^2 + (80-122)^2 + (158-122)^2 + (152-122)^2 + (161-122)^2 + (96-122)^2 + (122-122)^2 + (118-122)^2 + (87-122)^2 + (85-122)^2) / 12]
sd = sqrt[(169 + 1225 + 2354 + 1849 + 1296 + 144 + 1445 + 676 + 0 + 16 + 1225 + 1449) / 12]
sd = sqrt[12522 / 12]
sd = sqrt(1043.5)
sd ≈ 32.31
Therefore, within 3 standard deviations of the mean, all the values fall.