Question

Lesson App 2.5 What cholesterol levels are unhealthy for teen boys?

High levels of cholesterol in the blood increase the risk of heart disease. For teenage boys,
the distribution of blood cholesterol is approximately normal with mean μ = 151.6 milligrams
of cholesterol per deciliter of blood (mg/dl) and standard deviation o = 25 mg/dl.
1. About what proportion of teen boys have cholesterol levels less than 100 mg/dl?
2. Cholesterol levels of 200 or higher are considered high for teenagers. What percent of
teen boys have high cholesterol?
I
3. Cholesterol levels between 170 mg/dl and 200 mg/dl are considered borderline high for
teenagers. What percent of teen boys have borderline high cholesterol levels?

Answer 1

Answer:24

Explanation:

because the standard deviation

Answer 2

1. 1.92% of teenage boys

2.97.36% of teenage boys

3.20.32% of teenage boys

Explanation:

1. To find the proportion of teenage boys with cholesterol levels less than 100 mg/dL, we can use the z-score formula and then look up the corresponding z-score in a standard normal distribution table.

The formula for the z-score is:

Z=X−μσZ=σX−μ​

Where:

XX is the value you want to find the proportion for (100 mg/dL in this case).

μμ is the mean (average) cholesterol level (151.6 mg/dL).

σσ is the standard deviation (25 mg/dL).

Let's calculate the z-score:

Z=100−151.625=−51.625=−2.064Z=25100−151.6​=25−51.6​=−2.064

Now, we'll find the proportion of boys with cholesterol levels less than 100 mg/dL by looking up the z-score in a standard normal distribution table. The z-score of -2.064 corresponds to approximately 0.0192.

So, about 1.92% (0.0192 as a proportion) of teenage boys have cholesterol levels less than 100 mg/dL.

2. To find the percentage of teenage boys with high cholesterol levels (200 mg/dL or higher), we can use the z-score formula and then calculate the probability using the standard normal distribution table.

The formula for the z-score is:

Z=X−μσZ=σX−μ​

Where:

XX is the value we want to find the proportion for (200 mg/dL in this case).

μμ is the mean (average) cholesterol level (151.6 mg/dL).

σσ is the standard deviation (25 mg/dL).

Let's calculate the z-score:

Z=200−151.625=48.425=1.936Z=25200−151.6​=2548.4​=1.936

Now, we'll find the proportion of boys with cholesterol levels of 200 mg/dL or higher by looking up the z-score of 1.936 in the standard normal distribution table. The z-score of 1.936 corresponds to approximately 0.9736.

To find the percentage, we multiply this proportion by 100:

Percentage = 0.9736 * 100 ≈ 97.36%

So, approximately 97.36% of teenage boys have cholesterol levels that are considered high (200 mg/dL or higher).

3.To find the percentage of teenage boys with borderline high cholesterol levels (between 170 mg/dL and 200 mg/dL), we can use the z-score formula and calculate the probabilities for both values and then subtract to find the percentage in the range.

Calculate the z-scores for both values:

For 170 mg/dL:

Z1=170−151.625=18.425=0.736Z1​=25170−151.6​=2518.4​=0.736

For 200 mg/dL:

Z2=200−151.625=48.425=1.936Z2​=25200−151.6​=2548.4​=1.936

Find the probabilities for each z-score using the standard normal distribution table:

For Z1Z1​ (0.736), the corresponding probability is approximately 0.7704.

For Z2Z2​ (1.936), the corresponding probability is approximately 0.9736.

Calculate the difference in probabilities:

Difference = Probability at Z2Z2​ - Probability at Z1Z1​

Difference = 0.9736 - 0.7704 ≈ 0.2032

Convert the difference to a percentage:

Percentage = Difference * 100 ≈ 20.32%

So, approximately 20.32% of teenage boys have cholesterol levels in the borderline high range (between 170 mg/dL and 200 mg/dL).