Question

cigarette taxes the increases (in cents) in cigarette taxes for 15 states in a 6 - month period are 70, 60, 42, 20, 30, 31, 32, 34, 40, 45, 51, 50, 18, 40, 69
part 1 of 4
(a) find the range.
the range is 52.
part: 1 / 4
part 2 of 4
(b) find the variance and the standard deviation. assume the data represent samples, and use the shortcut formula for the unbiased estimator to compute the variance and standard deviation. round your answers to at least one decimal place.
the variance is 220.6
the standard deviation is 14.9.

Explanation:

Step1: List all sample data

Sample data: $18, 20, 30, 31, 32, 34, 40, 40, 42, 45, 50, 51, 60, 69, 70$; $n=15$

Step2: Calculate sample mean

$$\bar{x} = \frac{18+20+30+31+32+34+40+40+42+45+50+51+60+69+70}{15} = \frac{662}{15} \approx 44.133$$

Step3: Compute $\sum x^2$

$$\begin{align*} \sum x^2 &= 18^2+20^2+30^2+31^2+32^2+34^2+40^2+40^2+42^2+45^2+50^2+51^2+60^2+69^2+70^2 \\ &= 324+400+900+961+1024+1156+1600+1600+1764+2025+2500+2601+3600+4761+4900 \\ &= 30116 \end{align*}$$

Step4: Calculate sample variance

Use shortcut formula $s^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}$

$$\begin{align*} s^2 &= \frac{30116 - \frac{662^2}{15}}{15-1} \\ &= \frac{30116 - \frac{438244}{15}}{14} \\ &\approx \frac{30116 - 29216.2667}{14} \\ &\approx \frac{899.7333}{14} \\ &\approx 64.3 \end{align*}$$

Step5: Calculate sample standard deviation

$$s = \sqrt{s^2} \approx \sqrt{64.3} \approx 8.0$$

Answer:

The variance is $64.3$
The standard deviation is $8.0$