Answer to Question #336410 in Statistics and Probability for aurora

Question

1.      Find the range, the standard deviation, and the variance for the given samples. Round noninteger results to the nearest tenth.

       a. 1, 2, 5, 7, 8, 19, 22

       b. 3, 4, 7, 11, 12, 12, 15, 16

       c. 2.1, 3.0, 1.9, 1.5, 4.8

       d. 5.2, 11.7, 19.1, 3.7, 8.2, 16.3

       e. 48, 91, 87, 93, 59, 68, 92, 100, 81

Answer 1

a.

We have sample size n=7.

The smallest data value is 1 and the largest is 22. The range is:

$$22-1=21$$

Mean = 

$$\bar{x}=\dfrac{1+2+5+7+8+19+22}{7}=9.15$$

Variance

$$s^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n-1}=67.1$$

Standard deviation

$$s=\sqrt{s^2}=8.2$$

b.

We have sample size n=8.

The smallest data value is 3 and the largest is 16. The range is:

$$16-3=13$$

Mean = 

$$\bar{x}=\dfrac{3+4+7+11+12+12+15+16}{8}=10$$

Variance

$$s^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n-1}=23.4$$

Standard deviation

$$s=\sqrt{s^2}=4.8$$

c.

We have sample size n=5.

The smallest data value is 1.5 and the largest is 4.8. The range is:

$$4.8-1.5=3.3$$

Mean = 

$$\bar{x}=\dfrac{2.1+3.0+1.9+1.5+4.8}{5}=2.7$$

Variance

$$s^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n-1}=1.7$$

Standard deviation

$$s=\sqrt{s^2}=1.3$$

d.

We have sample size n=6.

The smallest data value is 3.7 and the largest is 19.1. The range is:

$$19.1-3.7=15.4$$

Mean = 

$$\bar{x}=\dfrac{5.2+11.7+19.1+3.7+8.2+16.3}{6}=10.7$$

Variance

$$s^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n-1}=37.7$$

Standard deviation

$$s=\sqrt{s^2}=6.1$$

e.

We have sample size n=9.

The smallest data value is 48 and the largest is 100. The range is:

$$100-48=52$$

Mean = 

$$\bar{x}=\dfrac{1}{9}(48+91+87+93+59$$

$$+68+92+100+81)=79.9$$

Variance

$$s^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n-1}=311.6$$

Standard deviation

$$s=\sqrt{s^2}=17.7$$