Solved on Dec 08, 2023

Compare the z-scores of the tallest (230230 cm) and shortest (136.6136.6 cm) men, given a mean of 175.78175.78 cm and standard deviation of 6.016.01 cm. The man with the more extreme z-score had the more extreme height.

STEP 1

Assumptions
1. The height of the tallest living man is 230 cm230 \mathrm{~cm}.
2. The height of the shortest living man is 136.6 cm136.6 \mathrm{~cm}.
3. The mean height of men at that time is 175.78 cm175.78 \mathrm{~cm}.
4. The standard deviation of men's heights at that time is 6.01 cm6.01 \mathrm{~cm}.
5. We are comparing the extremeness of the heights using zz scores.
6. zz scores are calculated using the formula z=xμσz = \frac{x - \mu}{\sigma}, where xx is the value being compared, μ\mu is the mean, and σ\sigma is the standard deviation.

STEP 2

Calculate the zz score for the tallest man using the formula for zz scores.
ztallest=xtallestμσz_{\text{tallest}} = \frac{x_{\text{tallest}} - \mu}{\sigma}

STEP 3

Plug in the values for the tallest man's height, the mean height, and the standard deviation to calculate his zz score.
ztallest=230175.786.01z_{\text{tallest}} = \frac{230 - 175.78}{6.01}

STEP 4

Perform the subtraction in the numerator.
ztallest=54.226.01z_{\text{tallest}} = \frac{54.22}{6.01}

STEP 5

Calculate the zz score for the tallest man.
ztallest=54.226.019.02z_{\text{tallest}} = \frac{54.22}{6.01} \approx 9.02

STEP 6

Round the zz score for the tallest man to two decimal places.
ztallest9.02z_{\text{tallest}} \approx 9.02

STEP 7

Calculate the zz score for the shortest man using the formula for zz scores.
zshortest=xshortestμσz_{\text{shortest}} = \frac{x_{\text{shortest}} - \mu}{\sigma}

STEP 8

Plug in the values for the shortest man's height, the mean height, and the standard deviation to calculate his zz score.
zshortest=136.6175.786.01z_{\text{shortest}} = \frac{136.6 - 175.78}{6.01}

STEP 9

Perform the subtraction in the numerator.
zshortest=39.186.01z_{\text{shortest}} = \frac{-39.18}{6.01}

STEP 10

Calculate the zz score for the shortest man.
zshortest=39.186.016.52z_{\text{shortest}} = \frac{-39.18}{6.01} \approx -6.52

STEP 11

Round the zz score for the shortest man to two decimal places.
zshortest6.52z_{\text{shortest}} \approx -6.52

STEP 12

Compare the absolute values of the zz scores to determine which man had the more extreme height.
ztallest=9.02=9.02|z_{\text{tallest}}| = |9.02| = 9.02 zshortest=6.52=6.52|z_{\text{shortest}}| = |-6.52| = 6.52

STEP 13

Since ztallest>zshortest|z_{\text{tallest}}| > |z_{\text{shortest}}|, the tallest man had the more extreme height.
The zz score for the tallest man is z=9.02z=9.02 and the zz score for the shortest man is z=6.52z=-6.52, the tallest man had the height that was more extreme.

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