you collected data on how many hours middle school students spent helping with chores each week in a local community. the responses from 10 students were: \{7, 9, 10, 12, 11, 9, 8, 13, 10, 11\}. using this data set that follows a normal distribution, what is the estimated population percentage of students helping chores between 4.8 and 15.2 hours per week?

90%
68%
99.7%
95%

Question

you collected data on how many hours middle school students spent helping with chores each week in a local community. the responses from 10 students were: \\{7, 9, 10, 12, 11, 9, 8, 13, 10, 11\\}. using this data set that follows a normal distribution, what is the estimated population percentage of students helping chores between 4.8 and 15.2 hours per week?

90%
68%
99.7%
95%

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", "Normal Distribution" ] } ## Calculate the sample mean $$ \bar{x} = \frac{7 + 9 + 10 + 12 + 11 + 9 + 8 + 13 + 10 + 11}{10} = \frac{100}{10} = 10 $$ ## Calculate the sample standard deviation $$ \begin{aligned} s &= \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \ &= \sqrt{\frac{(7-10)^2 + (9-10)^2 + (10-10)^2 + (12-10)^2 + (11-10)^2 + (9-10)^2 + (8-10)^2 + (13-10)^2 + (10-10)^2 + (11-10)^2}{9}} \ &= \sqrt{\frac{9 + 1 + 0 + 4 + 1 + 1 + 4 + 9 + 0 + 1}{9}} = \sqrt{\frac{30}{9}} \approx 1.8257 \end{aligned} $$ ## Apply the Empirical Rule $$ \begin{aligned} &\bar{x} - 3s \approx 10 - 3(1.8257) = 4.52 \ &\bar{x} - 2.85s \approx 4.8 \ &\bar{x} + 2.85s \approx 15.2 \ &\text{Using standard sample standard deviation } s = \sqrt{\frac{30}{9}} \approx 1.73 \text{ if using population formula } \sigma = \sqrt{\frac{30}{10}} = 1.732: \ &\mu - 3\sigma = 10 - 3(1.732) = 4.8 \ &\mu + 3\sigma = 10 + 3(1.732) = 15.2 \ &\text{Percentage within } 3 \text{ standard deviations} = 99.7\% \end{aligned} $$

90%

68%

99.7%

95%

{ "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

Answer: