1. Convert Ken's Measurements to Z-Scores

In recent years, there has been considerable discussion about the appropriateness of the body shapes and proportions of the Ken and Barbie dolls. Researchers investigating the dolls' body shapes scaled Ken and Barbie up to a common height of 170.18 cm (5'7") and compared them to body measurements of active adults. The table below presents measurements for Ken and Barbie and their reference groups.

| | Ken | Barbie |
|------------------------|---------------------------|----------------------------|
| | Chest Waist Hips | Chest Waist Hips |
| Doll | 75.0 56.5 72.0 | 82.3 40.7 72.7 |
| Human x-bar | 91.2 80.9 93.7 | 90.3 69.8 97.9 |
| Human S | 4.8 9.8 6.8 | 5.5 4.7 5.4 |

Convert Ken's chest, waist, and hips measurements to z-scores. Which of these measures appears to be the most different from Ken's reference group (human males)? Justify your response with an appropriate statistical argument.

2. Convert Barbie's Measurements to Z-Scores

Researchers scaled Ken and Barbie up to a common height and compared them to body measurements of active adults. Measurements for Ken and Barbie and their reference groups are shown in the table above.

Convert Barbie's chest, waist, and hips measurements to z-scores. Do these z-scores provide evidence to justify the claim that the Barbie doll is too thin of a representation of adult women? Justify your response with an appropriate statistical argument.

3. Standardized Scores for SAT and ACT

Jill scores 680 on the mathematics part of the SAT. The distribution of SAT scores in a reference population is normally distributed with a mean of 500 and a standard deviation of 100. Jack takes the ACT mathematics test and scores 27. ACT scores are normally distributed with a mean of 18 and a standard deviation of 6.

Find the standardized scores for both students.

4. Comparing SAT and ACT Scores

Assuming that both tests measure the same kind of ability, who has the higher score, and why?

5. Standard Deviations from Mean for Gross Income

A lunch stand in the business district has a mean daily gross income of $420 with a standard deviation of $50. Assume that the daily gross income is normally distributed.

If a randomly selected day has a gross income of $520, how many standard deviations away from the mean is that day's gross income?

1.)
zChest -> (91.2-75)/4.8=-3.4zWaist -> (80.9-56.5)/9.8≈-2.5zHips -> (93.7-72)/6.8≈-3.2It would be Ken's chest because the difference between the doll and an averaged human male is the greatest

2.)
zChest -> (82.3-90.3)/4.7≈-1.45zWaist -> (40.7-69.8)/4.7≈-6.19zHips -> (72.7-97.9)/5.4≈-4.7
Yes because the waist is far too thin compared to an averaged human woman

3.)
zJill -> (680-500)/100=1.8zJack -> (27-18)/6=1.5

4.)
It would be Jill because she has the higher z-score

5.)
$520-420=80$50=SDs$80(1 SD/50)=1.6 SD