Final answer:
The range of the cigarette tax increases is 62 cents. The variance and standard deviation would be calculated using the unbiased estimator formula for samples, and an estimated standard deviation can be roughly found by dividing the range by 4, giving 15.5 cents as an estimation.
Step-by-step explanation:
Calculating Range, Variance, and Standard Deviation
To calculate the range, we find the difference between the highest and the lowest increases in cigarette taxes. So, from the data provided (70, 12, 42, 45, 50, 51, 60, 69, 8, 40, 18), we have:
Range = Highest tax increase - Lowest tax increase = 70 - 8 = 62 cents.
For the variance and standard deviation, since the data is a sample, we'll use the unbiased estimator (n-1 in the denominator). We first find the mean of the data and then use the shortcut formula for each value:
Mean = Σx/n where x is each data point and n is the number of data points.
Variance (s^2) = Σ(x - Mean)^2 / (n - 1)
Standard Deviation (s) = √Variance
To estimate the standard deviation using the range rule of thumb, we take the range (62 cents for our data) and divide it by 4, as the rule of thumb suggests that the standard deviation is approximately the range/4.
The estimated standard deviation is therefore 62/4 = 15.5 cents.