you collected data on the test scores from a sample of students in a mathematics exam. the scores from 15 students were: 72, 78, 76, 74, 82, 80, 78, 76, 74, 82, 84, 86, 88, 90, 92. calculate the mean and standard deviation. then, using this data, estimate the percentage of students scoring within one standard deviation of the mean.

type your answer in the box. use the $\sqrt{x}$ button to enter math symbols and expressions. for decimal values, type your answer in two decimal places.

the mean of the data set is .

the standard deviation of the data set is .

this means about of the students score within one standard deviation of the mean, between and .

Question

you collected data on the test scores from a sample of students in a mathematics exam. the scores from 15 students were: 72, 78, 76, 74, 82, 80, 78, 76, 74, 82, 84, 86, 88, 90, 92. calculate the mean and standard deviation. then, using this data, estimate the percentage of students scoring within one standard deviation of the mean.

type your answer in the box. use the $\sqrt{x}$ button to enter math symbols and expressions. for decimal values, type your answer in two decimal places.

the mean of the data set is  .

the standard deviation of the data set is  .

this means about   of the students score within one standard deviation of the mean, between   and  .

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": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ], "new_concepts": [], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ] } ## Calculate the sample mean Using the *Sample Mean* knowledge point $$ \begin{aligned} n &= 15\ \sum x &= 72+78+76+74+82+80+78+76+74+82+84+86+88+90+92 = 1212\ \bar{x} &= \frac{1212}{15} = 80.8 \end{aligned} $$ ## Calculate the sample standard deviation Using the *Sample Standard Deviation* knowledge point $$ \begin{aligned} \sum x^2 &= 72^2+78^2+76^2+74^2+82^2+80^2+78^2+76^2+74^2+82^2+84^2+86^2+88^2+90^2+92^2\ &= 5184+6084+5776+5476+6724+6400+6084+5776+5476+6724+7056+7396+7744+8100+8464\ &= 98468\ s &= \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}} = \sqrt{\frac{98468 - \frac{1212^2}{15}}{14}} = \sqrt{\frac{98468 - 97929.6}{14}} = \sqrt{\frac{538.4}{14}} \approx 6.20 \end{aligned} $$ ## Determine the interval within one standard deviation Using the *Empirical Rule* knowledge point $$ \begin{aligned} \text{Lower bound} &= \bar{x} - s = 80.8 - 6.20 = 74.6\ \text{Upper bound} &= \bar{x} + s = 80.8 + 6.20 = 87.0 \end{aligned} $$ ## Calculate the percentage of students within the interval Using the *Empirical Rule* knowledge point $$ \begin{aligned} \text{Scores in } [74.6, 87.0] &: 78, 76, 82, 80, 78, 76, 82, 84, 86 \quad (\text{Count} = 9)\ \text{Percentage} &= \frac{9}{15} \times 100\% = 60\% \end{aligned} $$ You collected data on the test scores from a sample of students in a mathematics exam. The scores from 15 students were: 72, 78, 76, 74, 82, 80, 78, 76, 74, 82, 84, 86, 88, 90, 92. Calculate the mean and standard deviation. Then, using this data, estimate the percentage of students scoring within one standard deviation of the mean. Type your answer in the box. Use the $\sqrt{x}$ button to enter math symbols and expressions. For decimal values, type your answer in two decimal places. The mean of the data set is 80.8. The standard deviation of the data set is 6.20. This means about 60% of the students score within one standard deviation of the mean, between 74.6 and 87.0. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Sample Mean and Standard Deviation" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Determine the interval within one standard deviation

Calculate the percentage of students within the interval

Answer: