you collected data on how many minutes people spent walking their dogs each day. the responses from 15 individuals were: \{48, 50, 45, 47, 48, 49, 50, 47, 45, 49, 46, 48, 48, 50, 50\}.

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

the mean of the data is minutes, and the standard deviation is minutes.

about \% of the students spend between 44.65 and 51.35 hours per week volunteering in the community.

Question

you collected data on how many minutes people spent walking their dogs each day. the responses from 15 individuals were: \{48, 50, 45, 47, 48, 49, 50, 47, 45, 49, 46, 48, 48, 50, 50\}.

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

the mean of the data is   minutes, and the standard deviation is   minutes.

about  \% of the students spend between 44.65 and 51.35 hours per week volunteering in the community.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Sample Mean" ], "new_concepts": [ "Sample Standard Deviation", "Empirical Rule" ], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ] } ## Calculate the sample mean Using the **Sample Mean** knowledge point The dataset consists of $n = 15$ values: $$ \{48, 50, 45, 47, 48, 49, 50, 47, 45, 49, 46, 48, 48, 50, 50\} $$ Sum of the values: $$ \sum x = 48 + 50 + 45 + 47 + 48 + 49 + 50 + 47 + 45 + 49 + 46 + 48 + 48 + 50 + 50 = 720 $$ Mean: $$ \bar{x} = \frac{720}{15} = 48 $$ ## Calculate the sample standard deviation Find the squared differences from the mean $\bar{x} = 48$: $$ \begin{aligned} \sum (x - \bar{x})^2 = &(48-48)^2 + (50-48)^2 + (45-48)^2 + (47-48)^2 + (48-48)^2 \ &+ (49-48)^2 + (50-48)^2 + (47-48)^2 + (45-48)^2 + (49-48)^2 \ &+ (46-48)^2 + (48-48)^2 + (48-48)^2 + (50-48)^2 + (50-48)^2 \ = &0 + 4 + 9 + 1 + 0 + 1 + 4 + 1 + 9 + 1 + 4 + 0 + 0 + 4 + 4 = 42 \end{aligned} $$ Sample variance: $$ s^2 = \frac{42}{15 - 1} = \frac{42}{14} = 3 $$ Sample standard deviation: $$ s = \sqrt{3} \approx 1.73 $$ ## Determine the percentage within the interval The second question asks about the percentage of students spending between $44.65$ and $51.35$ hours. We calculate the number of standard deviations from the mean: $$ \begin{aligned} \text{Lower limit: } & 48 - k \cdot s = 44.65 \implies k \cdot 1.732 = 3.35 \implies k \approx 1.93 \approx 2 \ \text{Upper limit: } & 48 + k \cdot s = 51.35 \implies k \cdot 1.732 = 3.35 \implies k \approx 1.93 \approx 2 \end{aligned} $$ Using the Empirical Rule, approximately $95\%$ of the data falls within $2$ standard deviations of the mean. You collected data on how many minutes people spent walking their dogs each day. The responses from 15 individuals were: $\{48, 50, 45, 47, 48, 49, 50, 47, 45, 49, 46, 48, 48, 50, 50\}$. Type your answer in the box. Use the $\sqrt{x}$ button to enter math symbols and expressions. For values with decimals, type your answer in two decimal places. The mean of the data is 48 minutes, and the standard deviation is 1.73 minutes. About 95 % of the students spend between 44.65 and 51.35 hours per week volunteering in the community. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Sample Standard Deviation" ] }

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Question

Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Determine the percentage within the interval

Answer: