use the following distribution to complete parts (a) through (d) below.

1, 4, 7, 12, 11

a) compute the mean and standard deviation of the distribution.

the mean is 7.

the standard deviation is 4.64.
(round to the nearest hundredth as needed.)

b) multiply each number in the distribution by 3 and compute the mean and the standard deviation of this new distribution.

the mean is 21.

the standard deviation is .
(round to the nearest hundredth as needed.)

Question

use the following distribution to complete parts (a) through (d) below.

1, 4, 7, 12, 11

a) compute the mean and standard deviation of the distribution.

the mean is 7.

the standard deviation is 4.64.
(round to the nearest hundredth as needed.)

b) multiply each number in the distribution by 3 and compute the mean and the standard deviation of this new distribution.

the mean is 21.

the standard deviation is  .
(round to the nearest hundredth as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Mean Calculation", "Sample Standard Deviation", "Linear Transformations of Data" ], "new_concepts": [], "current_concepts": [ "Mean Calculation", "Sample Standard Deviation", "Linear Transformations of Data" ] } ## Calculate the original mean and standard deviation Using the *Mean Calculation* and *Sample Standard Deviation* knowledge points $$ \begin{aligned} &\text{Data: } 1, 4, 7, 12, 11 \quad (n = 5)\ &\bar{x} = \frac{1 + 4 + 7 + 12 + 11}{5} = \frac{35}{5} = 7\ &s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} = \sqrt{\frac{(1-7)^2 + (4-7)^2 + (7-7)^2 + (12-7)^2 + (11-7)^2}{4}}\ &s = \sqrt{\frac{36 + 9 + 0 + 25 + 16}{4}} = \sqrt{\frac{86}{4}} = \sqrt{21.5} \approx 4.64 \end{aligned} $$ ## Apply linear transformation to the distribution Using the *Linear Transformations of Data* knowledge point $$ \begin{aligned} &\text{New Data: } 3, 12, 21, 36, 33\ &\bar{x}_{\text{new}} = 3 \times \bar{x} = 3 \times 7 = 21\ &s_{\text{new}} = 3 \times s = 3 \times \sqrt{21.5} = \sqrt{193.5} \approx 13.91 \end{aligned} $$ b) Multiply each number in the distribution by 3 and compute the mean and the standard deviation of this new distribution. The mean is 21. The standard deviation is 13.91. (Round to the nearest hundredth as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Linear Transformations of Data" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the original mean and standard deviation

Apply linear transformation to the distribution

Answer: