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Q. 10.

Expert-verifiedFound in: Page 367

Book edition
6th

Author(s)
Daren Starnes, Josh Tabor

Pages
837 pages

ISBN
9781319113339

Working out Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many times did you go to an exercise or fitness center or work out?” Call the response Y for short. Based on a large sample survey, here is the probability distribution of Y

Part (a). A histogram of the probability distribution is shown. Describe its shape.

Part (b). Calculate and interpret the expected value of Y.

Part (a)

Rightward skewed

The most common number of days working out is unimodal 0

The number of days spent exercising ranges from 0 to 7.

Part (b) $\mu =1.03$

Here is the probability distribution of Y.

Because the highest bar in the histogram is to the left, and there is a tail of smaller bars to its right, the distribution is skewed to the right.

Because there is only one peak in the histogram, the distribution is unimodal.

Because the highest bar in the histogram is centred at 0, the most common number of days working out is 0.

The number of days spent exercising ranges from 0 to 7.

As a result:

Rightward skewed

The most common number of days working out is unimodal 0

The number of days spent exercising ranges from 0 to 7.

Days ${y}_{i}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Probability ${p}_{i}$ | 0.68 | 0.05 | 0.07 | 0.08 | 0.05 | 0.04 | 0.01 | 0.02 |

The expected value (or mean) is calculated by multiplying each possibility y by its probability ${p}_{i}$

$\mu =\sum _{}{y}_{i}{p}_{i}\phantom{\rule{0ex}{0ex}}\mu ==0x0.68+1x0.05+2x0.07+3x0.08+4x0.05+\phantom{\rule{0ex}{0ex}}5x0.04+6x0.01+7x0.02\phantom{\rule{0ex}{0ex}}\mu =1.03$

The average number of days spent working out is 1.03 days.

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