Bio Statistics question

Question

Suppose that the blood cholesterol levels of all men aged 20 to 34 years follows the Normal distribution with mean 𝜇=188 milligrams per deciliter (mg/dl) and standard deviation 𝜎=41

 mg/dl.

(a) Choose an SRS of 100 men from this population. What is the sampling distribution of 𝑥¯?

The sampling distribution of 𝑥¯

is 𝑁(188 mg/dl,4.1 mg/dl)

 .

𝑁(18.8 mg/dl,41 mg/dl)

 .

𝑁(18.8 mg/dl,4.1 mg/dl)

 .

𝑁(188 mg/dl,41 mg/dl)

 .

What is the probability that 𝑥¯takes a value between 185 and 191 mg/dl? This is the probability that 𝑥¯ estimates 𝜇 within ±3 mg/dl. (Enter your answer rounded to four decimal places.)

𝑃(185<𝑥¯<191)𝑛=100=

(b) Choose an SRS of 1000

men from this population. What is the sampling distribution of 𝑥¯?

The sampling distribution of 𝑥¯ is

𝑁(188 mg/dl,0.0041 mg/dl)

 .

𝑁(188 mg/dl,4.1 mg/dl)

 .

𝑁(18.8 mg/dl,1.2965 mg/dl)

 .

𝑁(188 mg/dl,1.2965 mg/dl)

 .

Now what is the probability that 𝑥¯ falls within ±3 mg/dl of 𝜇 ? The larger sample is much more likely to give an accurate estimate of 𝜇 . (Enter your answer rounded to four decimal places.)

𝑃(185<𝑥¯<191)𝑛=1000=

Answer 1

Central Limit Theorem allows us to make probability statements about the sample mean, specifically in relation to its value in comparison to the population mean,  For example, the mean of the measurements in a sample of size n where the distribution of 𝑋 is its sampling distribution, with mean 𝜇𝑋=𝜇 and a standard deviation 𝜎𝑋=√σ⁄√n. You just need to remember to use the population distribution and the size (n) of your sample to calculate the µ and σ of any size sample drawn from it. Looks like your taking a sample of 100 (n) in the first problem, and of size=1000 in the second one, For all your questions about finding the sample mean, just use the mean of the population, which is given as 118.0 mg/dl, and to find the σ, just divide the population σ by the √ of n. That's how you get the answer to your multiple choice questions. Make sure you sue the correct 'n', they're different in your 2 questions.

For the other two questions, it looks like they set up the equation you need to solve for you, and you just learned that the variable you're working with is also normal, and you also found the sample µ and σ of the variable, so you can solve those equations just like you would any 'regular' normally distributed variable, i.e. standardize the variables. For example:

𝑃(110 < 𝑋 < 114) = 𝑃((110−𝜇𝑋)/𝜎𝑋 < 𝑍 < (114−𝜇𝑋)/⎯𝜎𝑋).

I need to adhere to guidelines & can't do the actual 'solving' for you, so I'll leave the calculations to you. Good luck!