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.

Explanation:

Calculate the sample mean

We first sum the 15 data values:

$$ \sum x = 48 + 50 + 45 + 47 + 48 + 49 + 50 + 47 + 45 + 49 + 46 + 48 + 48 + 50 + 50 = 720 $$

The sample size is \(n = 15\).
The mean \(\bar{x}\) is:

$$ \bar{x} = \frac{720}{15} = 48 $$

Calculate the sample standard deviation

We compute the squared deviations from the mean \(\bar{x} = 48\):

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Using the sample standard deviation formula:

$$ s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} = \sqrt{\frac{42}{14}} = \sqrt{3} \approx 1.73 $$

Analyze the second question

The second question asks: "About [ ]% of the students spend between 44.65 and 51.35 hours per week volunteering in the community."
We check the interval boundaries relative to the mean \(\bar{x} = 48\) and standard deviation \(s = 1.73\):

$$ \frac{51.35 - 48}{1.732} \approx \frac{3.35}{1.732} \approx 1.93 \approx 2 $$

$$ \frac{44.65 - 48}{1.732} \approx \frac{-3.35}{1.732} \approx -1.93 \approx -2 $$

The interval \([44.65, 51.35]\) represents approximately 2 standard deviations from the mean (\(\bar{x} \pm 2s\)).
According to the Empirical Rule (68-95-99.7 Rule), approximately 95% of the data falls within 2 standard deviations of the mean.

Answer:

Question 1

The mean of the data is <blank>48</blank> minutes, and the standard deviation is <blank>1.73</blank> minutes.

Question 2

About <blank>95</blank>% of the students spend between 44.65 and 51.35 hours per week volunteering in the community.