you collected data on how many hours students spend on their favorite leisure activities each week. the responses from 15 students are as follows:

\{60, 62, 64, 65, 67, 63, 66, 64, 68, 66, 63, 65, 62, 67, 64\}

given the following data set that fits a normal distribution:

* calculate the mean and standard deviation.
* then, estimate the population percentage within 3 standard deviations of the mean.

drop down the options and select the correct values.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 99.7% of the data falls between dropdown and dropdown

Question

you collected data on how many hours students spend on their favorite leisure activities each week. the responses from 15 students are as follows:

\{60, 62, 64, 65, 67, 63, 66, 64, 68, 66, 63, 65, 62, 67, 64\}

given the following data set that fits a normal distribution:

* calculate the mean and standard deviation.
* then, estimate the population percentage within 3 standard deviations of the mean.

drop down the options and select the correct values.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 99.7% 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": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ], "new_concepts": [], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ] } ## Calculate the sample mean $$ \begin{aligned} n &= 15 \ \sum x &= 60 + 62 + 64 + 65 + 67 + 63 + 66 + 64 + 68 + 66 + 63 + 65 + 62 + 67 + 64 = 966 \ \bar{x} &= \frac{966}{15} = 64.4 \end{aligned} $$ ## Calculate the sample standard deviation $$ \begin{aligned} \sum (x - \bar{x})^2 &= (60-64.4)^2 + 2(62-64.4)^2 + 2(63-64.4)^2 + 3(64-64.4)^2 \ &\quad + 2(65-64.4)^2 + 2(66-64.4)^2 + 2(67-64.4)^2 + (68-64.4)^2 \ &= 19.36 + 2(5.76) + 2(1.96) + 3(0.16) + 2(0.36) + 2(2.56) + 2(6.76) + 12.96 \ &= 19.36 + 11.52 + 3.92 + 0.48 + 0.72 + 5.12 + 13.52 + 12.96 = 67.6 \ s &= \sqrt{\frac{67.6}{15-1}} = \sqrt{\frac{67.6}{14}} \approx \sqrt{4.82857} \approx 2.2 \end{aligned} $$ ## Determine the Empirical Rule interval $$ \begin{aligned} \text{Lower Bound} &= \bar{x} - 3s = 64.4 - 3(2.2) = 64.4 - 6.6 = 57.8 \ \text{Upper Bound} &= \bar{x} + 3s = 64.4 + 3(2.2) = 64.4 + 6.6 = 71.0 \end{aligned} $$ The mean of the data set is 64.4 The standard deviation of the data set is 2.2 This means 99.7% of the data falls between 57.8 and 71 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Determine the Empirical Rule interval

Answer: