Question

The mean cholesterol levels of women age 45-59 in Ghana, Nigeria, and Seychelles is 5.1 mmol/l and the standard deviation is 1.0 mmol/l (Lawes, Hoorn, Law & Rodgers, 2004). Assume that cholesterol levels are normally distributed.

1. Find the probability that a woman age 45-59 in Ghana has a cholesterol level above 6.2 mmol/l (considered a high level).
2. Suppose doctors decide to test the woman’s cholesterol level again and average the two values.  Find the probability that this woman’s mean cholesterol level for the two tests is above 6.2 mmol/l. <----Which two numbers am I averaging?
3. Suppose doctors being very conservative decide to test the woman’s cholesterol level a third time and average the three values.  Find the probability that this woman’s mean cholesterol level for the three tests is above 6.2 mmol/l. <----Not sure which numbers I'm averaging?
4. If the sample mean cholesterol level for this woman after three tests is above 6.2 mmol/l, what could you conclude?

I'm answering these questions using the "normalcdf" function in my TI-84. I'm not sure how to get the new standard deviation/numbers I'm averaging for 2 and 3.

Answer 1

Given and

The probability that a woman age 45-59 in Ghana has a cholesterol level above 6.2 mmol/l (considered a high level) is 0.1357.

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For next two parts, recall central limit theorem.

"The Central limit theorem states that the sampling distribution of the sample mean is approximately normally distributed with mean μ and standard deviation σ/√n if either n is large or population is normal."

The standard deviation for woman’s mean cholesterol level for the two tests, , is:

In calculator, we can directly enter this expression (screenshot attached) in place of standard deviation to avoid round off error.

The probability that this woman’s mean cholesterol level for the two tests is above 6.2 mmol/l is 0.0599.

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The standard deviation for woman’s mean cholesterol level for the two tests, , is:

In calculator, we can directly enter this expression (screenshot attached) in place of standard deviation to avoid round off error.

The probability that this woman’s mean cholesterol level for the three tests is above 6.2 mmol/l is 0.0284.