Question

sabine records the daily heights of a random sample of bamboo stalks, in inches. they are:

20, 19, 17, 16, 18, 15, 20, 21

consider the formulas:

a. (s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1})

b. (s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}})

c. (\sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n})

d. (\sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}})

which formula should you use for variance?

which formula should you use for standard deviation?

Explanation:

Identify data type

Using the Population Variance Calculation and Population Standard Deviation knowledge points
The problem states that Sabine records the daily heights of a "random sample" of bamboo stalks. Since this dataset represents a sample rather than an entire population, we must use the sample formulas for variance and standard deviation.

Select variance formula

Using the Population Variance Calculation knowledge point
For a sample, the variance is denoted by \(s^2\) and uses a denominator of \(n-1\):

$$s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n-1}$$

This corresponds to formula A.

Select standard deviation formula

Using the Population Standard Deviation knowledge point
For a sample, the standard deviation is denoted by \(s\) and is the square root of the sample variance:

$$s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n-1}}$$

This corresponds to formula B.

Answer:

Question 1

Which formula should you use for variance? <blank>A</blank>

Question 2

Which formula should you use for standard deviation? <blank>B</blank>