The serum cholesterol levels for men in in one age group are normally distributed with mean of 178.1 and standard deviation of 40.8. All units are in mg/100ml. Find two levels that cut off the top 10% and the bottom 10 % from the distribution of x.
[178.1-1.28(40.8), 178.1+ 1.28(40.8) ] =[ ] ?
125.876,218.9?
125.9, 230.3
To find the two cholesterol levels that cut off the top 10% and the bottom 10% from the distribution, we need to calculate the corresponding percentiles of the data.
Step 1: Find the z-score for the top 10%.
The top 10% corresponds to the 90th percentile. To find the z-score for this percentile, we use the standard normal distribution table (also known as the z-table) or a calculator.
The formula to calculate the z-score is: z = (x - μ) / σ
where:
z is the z-score
x is the value of interest (unknown in this case)
μ is the mean of the distribution (178.1)
σ is the standard deviation of the distribution (40.8)
Using the z-table or calculator, we find that the z-score corresponding to the 90th percentile is approximately 1.28.
Step 2: Calculate the value for the top 10%.
Now that we have the z-score, we can use it to find the corresponding value (x) in the distribution.
1.28 = (x - 178.1) / 40.8
Solving this equation for x, we get:
x - 178.1 = 1.28 * 40.8
x - 178.1 = 52.224
x ≈ 230.324
Therefore, the cholesterol level that cuts off the top 10% from the distribution is approximately 230.324 mg/100ml.
Step 3: Find the z-score for the bottom 10%.
The bottom 10% corresponds to the 10th percentile. To find the z-score for this percentile, we use the same process as in Step 1.
Using the z-table or calculator, we find that the z-score corresponding to the 10th percentile is approximately -1.28 (since it's the opposite direction compared to the top 10%).
Step 4: Calculate the value for the bottom 10%.
Using the z-score (-1.28), we can find the corresponding value (x) in the distribution.
-1.28 = (x - 178.1) / 40.8
Solving this equation for x, we get:
x - 178.1 = -1.28 * 40.8
x - 178.1 = -52.224
x ≈ 125.876
Therefore, the cholesterol level that cuts off the bottom 10% from the distribution is approximately 125.876 mg/100ml.
To summarize:
- The cholesterol level that cuts off the top 10% from the distribution is approximately 230.324 mg/100ml.
- The cholesterol level that cuts off the bottom 10% from the distribution is approximately 125.876 mg/100ml.