given the following data set: 16, 18, 19, 20, 21, 22, 23, 23, 24, 25, which fits a normal distribution, calculate the mean and standard deviation. then, estimate the population percentage within 2 standard deviations of the mean.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 95% of the data falls between dropdown and dropdown

Question

given the following data set: 16, 18, 19, 20, 21, 22, 23, 23, 24, 25, which fits a normal distribution, calculate the mean and standard deviation. then, estimate the population percentage within 2 standard deviations of the mean.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 95% of the data falls between dropdown and dropdown

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ] } ## Calculate the sample mean $$ \begin{aligned} n &= 11 \ \sum x &= 16 + 18 + 19 + 20 + 21 + 22 + 23 + 23 + 24 + 25 + 21 = 232 \ \bar{x} &= \frac{232}{11} \approx 21.09 \end{aligned} $$ ## Calculate the sample standard deviation $$ \begin{aligned} \sum x^2 &= 16^2 + 18^2 + 19^2 + 20^2 + 21^2 + 22^2 + 23^2 + 23^2 + 24^2 + 25^2 + 21^2 \ &= 256 + 324 + 361 + 400 + 441 + 484 + 529 + 529 + 576 + 625 + 441 = 4966 \ s &= \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n - 1}} = \sqrt{\frac{4966 - \frac{232^2}{11}}{10}} = \sqrt{\frac{4966 - 4893.0909}{10}} \approx 2.70 \end{aligned} $$ ## Apply the Empirical Rule for 95% interval $$ \begin{aligned} \text{Lower Bound} &= \bar{x} - 2s = 21.09 - 2(2.70) = 15.69 \ \text{Upper Bound} &= \bar{x} + 2s = 21.09 + 2(2.70) = 26.49 \end{aligned} $$ Given the following data set: 16, 18, 19, 20, 21, 22, 23, 23, 24, 25, which fits a normal distribution, calculate the mean and standard deviation. Then, estimate the population percentage within 2 standard deviations of the mean. The mean of the data set is 21.09 The standard deviation of the data set is 2.70 This means 95% of the data falls between 15.69 and 26.49 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Apply the Empirical Rule for 95% interval

Answer: