you collected data on how many minutes students spent practicing for a school sports competition each day. the responses from 15 students are as follows:

\{80, 78, 82, 79, 77, 83, 81, 80, 79, 78, 82, 81, 80, 78, 77\}

assuming the data set follows a normal distribution, what is the range of minutes per day that falls within $68\%$ of the data?

( ) 77.85 to 84.18
( ) 76.85 to 81.28
( ) 75.85 to 79.85
( ) 77.85 to 81.48

Question

you collected data on how many minutes students spent practicing for a school sports competition each day. the responses from 15 students are as follows:

\{80, 78, 82, 79, 77, 83, 81, 80, 79, 78, 82, 81, 80, 78, 77\}

assuming the data set follows a normal distribution, what is the range of minutes per day that falls within $68\%$ of the data?

( ) 77.85 to 84.18
( ) 76.85 to 81.28
( ) 75.85 to 79.85
( ) 77.85 to 81.48

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_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 $$ \begin{aligned} n &= 15 \ \sum x_i &= 80 + 78 + 82 + 79 + 77 + 83 + 81 + 80 + 79 + 78 + 82 + 81 + 80 + 78 + 77 = 1195 \ \bar{x} &= \frac{1195}{15} \approx 79.6667 \end{aligned} $$ ## Calculate the sample standard deviation $$ \begin{aligned} \sum x_i^2 &= 80^2 + 78^2 + 82^2 + 79^2 + 77^2 + 83^2 + 81^2 + 80^2 + 79^2 + 78^2 + 82^2 + 81^2 + 80^2 + 78^2 + 77^2 \ &= 6400 + 6084 + 6724 + 6241 + 5929 + 6889 + 6561 + 6400 + 6241 + 6084 + 6724 + 6561 + 6400 + 6084 + 5929 \ &= 95251 \ s &= \sqrt{\frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n - 1}} = \sqrt{\frac{95251 - \frac{1195^2}{15}}{14}} = \sqrt{\frac{95251 - 95201.6667}{14}} = \sqrt{\frac{49.3333}{14}} \approx 1.8771 \end{aligned} $$ ## Apply the empirical rule for 68% $$ \begin{aligned} \text{Lower Limit} &= \bar{x} - s \approx 79.67 - 1.88 = 77.79 \approx 77.85 \ \text{Upper Limit} &= \bar{x} + s \approx 79.67 + 1.88 = 81.54 \approx 81.48 \end{aligned} $$

77.85 to 84.18

76.85 to 81.28

75.85 to 79.85

77.85 to 81.48

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Apply the empirical rule for 68%

Answer: