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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. [tex]\( s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n-1} \)[/tex]
B. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n-1}} \)[/tex]
C. [tex]\( \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
D. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]

Which formula should you use for variance? [tex]\( \square \)[/tex]

Which formula should you use for standard deviation? [tex]\( \square \)[/tex]

Sagot :

GinnyAnsweridn
To solve this problem, we need to identify the correct formulas for calculating the variance and standard deviation based on the given sample data.

### Step 1: Define the Data
We are given the daily heights of bamboo stalks:
[tex]\[ 20, 19, 17, 16, 18, 15, 20, 21 \][/tex]

### Step 2: Calculate the Mean ([tex]\(\bar{x}\)[/tex])
The mean ([tex]\(\bar{x}\)[/tex]) is calculated as:
[tex]\[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \][/tex]
where [tex]\( x_i \)[/tex] are the individual heights and [tex]\( n \)[/tex] is the number of observations.

For the given data:
[tex]\[ \bar{x} = \frac{20 + 19 + 17 + 16 + 18 + 15 + 20 + 21}{8} = \frac{146}{8} = 18.25 \][/tex]

### Step 3: Calculate the Sum of Squared Deviations
We calculate the squared deviation of each observation from the mean:
[tex]\[ (x - \bar{x})^2 \][/tex]

For each height:
[tex]\[ (20 - 18.25)^2 = 3.0625 \][/tex]
[tex]\[ (19 - 18.25)^2 = 0.5625 \][/tex]
[tex]\[ (17 - 18.25)^2 = 1.5625 \][/tex]
[tex]\[ (16 - 18.25)^2 = 5.0625 \][/tex]
[tex]\[ (18 - 18.25)^2 = 0.0625 \][/tex]
[tex]\[ (15 - 18.25)^2 = 10.5625 \][/tex]
[tex]\[ (20 - 18.25)^2 = 3.0625 \][/tex]
[tex]\[ (21 - 18.25)^2 = 7.5625 \][/tex]

Summing these squared deviations:
[tex]\[ 3.0625 + 0.5625 + 1.5625 + 5.0625 + 0.0625 + 10.5625 + 3.0625 + 7.5625 = 31.5 \][/tex]

### Step 4: Calculate the Variance
Using formula A:
[tex]\[ s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]

For the data:
[tex]\[ s^2 = \frac{31.5}{8 - 1} = \frac{31.5}{7} = 4.5 \][/tex]

### Step 5: Calculate the Standard Deviation
Using formula B:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]

For the data:
[tex]\[ s = \sqrt{4.5} \approx 2.1213 \][/tex]

### Conclusion
The variance of the sample is given by formula A:
[tex]\[ s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]

The standard deviation of the sample is given by formula B:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]

So, the formulas we should use are:

- For variance: Formula A [tex]\( \boxed{A} \)[/tex]
- For standard deviation: Formula B [tex]\( \boxed{B} \)[/tex]
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