Health High School

Too much cholesterol in the blood increases the risk of heart disease. The cholesterol level for young women is N(185,39) while the cholesterol level for middle-aged men is N(222,37). Level of 240 or more is consider high while level of 200-240 is borderline high. What fractions of the young women and middle-aged menhave high and border line levels?

Answers

Answer 1

Young Women: - Borderline high level: 0.057

- High level: 0.0785

Middle-aged Men:

- Borderline high level: 0.0895

- High level: 0.3121

To find the fraction of young women with a borderline high cholesterol level, we first calculate the z-scores for 200 and 240 using the mean (185) and standard deviation (39) provided:

z-score for 200 = (200 - 185) / 39 = 0.38 (approx)

z-score for 240 = (240 - 185) / 39 = 1.41 (approx)

Next, we look up the corresponding areas under the standard normal curve from the z-table:

P(0.38 < Z < 1.41) = 0.4090 (from the z-table)

P(Z < 0.38) = 0.3520 (from the z-table)

To find the fraction of young women with a borderline high level, we subtract the area from the z-score 1.41 (P(0.38 < Z < 1.41)) by the area from the z-score 0.38 (P(Z < 0.38)):

Fraction of young women with borderline high level = 0.4090 - 0.3520 = 0.057

For the fraction of young women with a high cholesterol level (240 or more), we use the z-score for 240:

z-score for 240 = (240 - 185) / 39 = 1.41 (approx)

From the z-table, we find the area under the standard normal curve:

P(Z > 1.41) = 0.0785 (from the z-table)

Therefore, the fraction of young women with a high level is 0.0785.

Moving on to middle-aged men, given their cholesterol level follows a normal distribution with a mean of 222 and standard deviation of 37:

To find the fraction of middle-aged men with a borderline high level (200-240), we calculate the z-scores:

z-score for 200 = (200 - 222) / 37 = -0.59 (approx)

z-score for 240 = (240 - 222) / 37 = 0.49 (approx)

Looking up the areas from the z-table:

P(-0.59 < Z < 0.49) = 0.3085 (from the z-table)

P(Z < -0.59) = 0.2190 (from the z-table)

By subtracting these areas, we find the fraction of middle-aged men with a borderline high level:

Fraction of middle-aged men with borderline high level = 0.3085 - 0.2190 = 0.0895

For the fraction of middle-aged men with a high cholesterol level (240 or more):

z-score for 240 = (240 - 222) / 37 = 0.49 (approx)

From the z-table:

P(Z > 0.49) = 0.3121 (from the z-table)

Therefore, the fraction of middle-aged men with a high level is 0.3121.

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