Suppose that the cholesterol level of women at ages 20-29 is normally distribute

Question

Suppose that the cholesterol level of women at ages 20-29 is normally distributed with a mean level 180 (in mg/dL) and standard deviation 35 (in mg/dL), Let X be the cholesterol level of a randomly selected women at ages 20-29. (a) Find the probability that the cholesterol level of the randomly selected woman is more than 195. (b) Suppose 16 women are picked at random. Let R be the mean cholesterol level of them. Find the probability that the sample mean cholesterol level is more than 195. (c) A new study is being planned which requires a group of 20-29 year old women representing the middle 70%. Determine the lower and upper cholesterol levels corresponding to this desired range. (d) In a final study of 20-29 year old women, researchers wish to validate the value of the population mean with a 95% confidence interval and margin of error greater than 5. Determine the minimum sample size for this requirement.

Explanation / Answer

a) here P(X>195)=1-P(X<195)=1-P(Z<(195-180)/35)=1-P(Z<0.4285)=1-0.6659=0.3341

b)for n=16; std error of mean =std deviaiton/(n)1/2 =8.75

P(X>195)=1-P(X<195)=1-P(Z<(195-180)/8.75)=1-P(Z<1.7143)=1-0.9568=0.0432

c)for middle 70% ; z=1.0364

hence interval =mean -/+z*std deviation =143.7248 ; 216.2752

d)for 95% CI, z=1.96

margin of errror E=5

hence sample size n=(z*std deviation/E)2 =~189

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