Question

Cigarette Taxes The increases (in cents) in cigarette taxes for 17 states in a 6-month period are 60, 20, 40, 40, 45, 12, 34, 51, 30, 70, 42, 31, 69, 32, 8, 18.50 )g Find the range, variance, and standard deviation for the data. Use the range rule of thumb to estimate the stand- ard deviation. Compare the estimate to the actual stand- ard deviation.

Answer 1

Range: 70 - 8 = 62

Variance: To calculate the variance, first find the mean (average) of the data. Then subtract the mean from each data point, square the differences, and average the squared differences.

Mean: (60 + 20 + 40 + 40 + 45 + 12 + 34 + 51 + 30 + 70 + 42 + 31 + 69 + 32 + 8 + 18.50) / 17 = 40.47

Variance: ( (60-40.47)^2 + (20-40.47)^2 + (40-40.47)^2 + (40-40.47)^2 + (45-40.47)^2 + (12-40.47)^2 + (34-40.47)^2 + (51-40.47)^2 + (30-40.47)^2 + (70-40.47)^2 + (42-40.47)^2 + (31-40.47)^2 + (69-40.47)^2 + (32-40.47)^2 + (8-40.47)^2 + (18.50-40.47)^2 ) / 17 = 517.41

Standard deviation: The standard deviation is the square root of the variance. So, the standard deviation is sqrt(517.41) = 22.79

Range rule of thumb estimate: The range rule of thumb estimate for the standard deviation is the range divided by 4. So, the estimate is 62 / 4 = 15.5

Comparison: The range rule of thumb estimate is 15.5, while the actual standard deviation is 22.79. The estimate is not very accurate for this data set, as it is much lower than the actual standard deviation