Solved on Dec 08, 2023

Compare the z-scores of the tallest ( $230$ cm) and shortest ( $136.6$ cm) men, given a mean of $175.78$ cm and standard deviation of $6.01$ 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 $230 \mathrm{~cm}$ .
2. The height of the shortest living man is $136.6 \mathrm{~cm}$ .
3. The mean height of men at that time is $175.78 \mathrm{~cm}$ .
4. The standard deviation of men's heights at that time is $6.01 \mathrm{~cm}$ .
5. We are comparing the extremeness of the heights using $z$ scores.
6. $z$ scores are calculated using the formula $z = \frac{x - \mu}{\sigma}$ , where $x$ is the value being compared, $\mu$ is the mean, and $\sigma$ is the standard deviation.

STEP 2

Calculate the $z$ score for the tallest man using the formula for $z$ scores.
$z_{\text{tallest}} = \frac{x_{\text{tallest}} - \mu}{\sigma}$

STEP 3

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

STEP 4

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

STEP 5

Calculate the $z$ score for the tallest man.
$z_{\text{tallest}} = \frac{54.22}{6.01} \approx 9.02$

STEP 6

Round the $z$ score for the tallest man to two decimal places.
$z_{\text{tallest}} \approx 9.02$

STEP 7

Calculate the $z$ score for the shortest man using the formula for $z$ scores.
$z_{\text{shortest}} = \frac{x_{\text{shortest}} - \mu}{\sigma}$

STEP 8

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

STEP 9

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

STEP 10

Calculate the $z$ score for the shortest man.
$z_{\text{shortest}} = \frac{-39.18}{6.01} \approx -6.52$

STEP 11

Round the $z$ score for the shortest man to two decimal places.
$z_{\text{shortest}} \approx -6.52$

STEP 12

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

STEP 13

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

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Compare the z-scores of the tallest ( $230$ cm) and shortest ( $136.6$ cm) men, given a mean of $175.78$ cm and standard deviation of $6.01$ cm. The man with the more extreme z-score had the more extreme height.

STEP 1

STEP 2

Calculate the $z$ score for the tallest man using the formula for $z$ scores.
$z_{\text{tallest}} = \frac{x_{\text{tallest}} - \mu}{\sigma}$

STEP 3

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

STEP 4

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

STEP 5

Calculate the $z$ score for the tallest man.
$z_{\text{tallest}} = \frac{54.22}{6.01} \approx 9.02$

STEP 6

Round the $z$ score for the tallest man to two decimal places.
$z_{\text{tallest}} \approx 9.02$

STEP 7

Calculate the $z$ score for the shortest man using the formula for $z$ scores.
$z_{\text{shortest}} = \frac{x_{\text{shortest}} - \mu}{\sigma}$

STEP 8

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

STEP 9

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

STEP 10

Calculate the $z$ score for the shortest man.
$z_{\text{shortest}} = \frac{-39.18}{6.01} \approx -6.52$

STEP 11

Round the $z$ score for the shortest man to two decimal places.
$z_{\text{shortest}} \approx -6.52$

STEP 12

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

STEP 13

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

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Compare the z-scores of the tallest (230230230 cm) and shortest (136.6136.6136.6 cm) men, given a mean of 175.78175.78175.78 cm and standard deviation of 6.016.016.01 cm. The man with the more extreme z-score had the more extreme height.

STEP 1

STEP 2

Calculate the zzz score for the tallest man using the formula for zzz scores.ztallest=xtallest−μσz_{\text{tallest}} = \frac{x_{\text{tallest}} - \mu}{\sigma}ztallest​=σxtallest​−μ​

STEP 3

Plug in the values for the tallest man's height, the mean height, and the standard deviation to calculate his zzz score.ztallest=230−175.786.01z_{\text{tallest}} = \frac{230 - 175.78}{6.01}ztallest​=6.01230−175.78​

STEP 4

Perform the subtraction in the numerator.ztallest=54.226.01z_{\text{tallest}} = \frac{54.22}{6.01}ztallest​=6.0154.22​

STEP 5

Calculate the zzz score for the tallest man.ztallest=54.226.01≈9.02z_{\text{tallest}} = \frac{54.22}{6.01} \approx 9.02ztallest​=6.0154.22​≈9.02

STEP 6

Round the zzz score for the tallest man to two decimal places.ztallest≈9.02z_{\text{tallest}} \approx 9.02ztallest​≈9.02

STEP 7

Calculate the zzz score for the shortest man using the formula for zzz scores.zshortest=xshortest−μσz_{\text{shortest}} = \frac{x_{\text{shortest}} - \mu}{\sigma}zshortest​=σxshortest​−μ​

STEP 8

Plug in the values for the shortest man's height, the mean height, and the standard deviation to calculate his zzz score.zshortest=136.6−175.786.01z_{\text{shortest}} = \frac{136.6 - 175.78}{6.01}zshortest​=6.01136.6−175.78​

STEP 9

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

STEP 10

Calculate the zzz score for the shortest man.zshortest=−39.186.01≈−6.52z_{\text{shortest}} = \frac{-39.18}{6.01} \approx -6.52zshortest​=6.01−39.18​≈−6.52

STEP 11

Round the zzz score for the shortest man to two decimal places.zshortest≈−6.52z_{\text{shortest}} \approx -6.52zshortest​≈−6.52

STEP 12

STEP 13

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Compare the z-scores of the tallest ( $230$ cm) and shortest ( $136.6$ cm) men, given a mean of $175.78$ cm and standard deviation of $6.01$ cm. The man with the more extreme z-score had the more extreme height.

Calculate the $z$ score for the tallest man using the formula for $z$ scores.
$z_{\text{tallest}} = \frac{x_{\text{tallest}} - \mu}{\sigma}$

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

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

Calculate the $z$ score for the tallest man.
$z_{\text{tallest}} = \frac{54.22}{6.01} \approx 9.02$

Round the $z$ score for the tallest man to two decimal places.
$z_{\text{tallest}} \approx 9.02$

Calculate the $z$ score for the shortest man using the formula for $z$ scores.
$z_{\text{shortest}} = \frac{x_{\text{shortest}} - \mu}{\sigma}$

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

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

Calculate the $z$ score for the shortest man.
$z_{\text{shortest}} = \frac{-39.18}{6.01} \approx -6.52$

Round the $z$ score for the shortest man to two decimal places.
$z_{\text{shortest}} \approx -6.52$