Answered

Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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 :

GinnyAnswer
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.