Question 1. A nutritionist thinks the average person with an income below the po

Question

Question 1. A nutritionist thinks the average person with an income below the poverty level gets less than the recommended daily allowance (RDA) of 800 mg of calcium. To test her conjecture, she obtains the daily intakes of calcium for a random sample of 35 people with incomes below the poverty level. The mean intake was found to be 747.3 mg. Assume that the standard deviation is "known" to be 2622 mg· +' a) Provide a 99% CI on the true mean RDA for individuals below the poverty level. b) If it was desired that the 99% CI does not exceed a width of 20 mg, how many people should have been used in the study? ^ c) At the 5% significance level, do the data provide sufficient evidence to conclude that the mean calcium intake of all people with incomes below the poverty level is less than the RDA of 800 mg?-' d) Provide the P-value for the test in part c) above. e) Up until now, we have assumed the standard deviation to be "known" but suppose it actually wasn't. In fact, it was actually determined that s = 2622mg is the estimated standard deviation based on the same n - 35 people. -' a. What is the 95% CI for the true standard deviation? +' b. Redo the test from part c), allowing for the fact that the standard deviation is estimated and not known.

Explanation / Answer

a) we know that the confidence interval is given as

mean +- z*sd/sqrt(n) , where z = 2.58 for 99% CI

mean = 747.3 n = 35 , sd = 262.2 , putting the values

747.3 +- 2.58*262.2/sqrt(35)

632.95 , 861.64

b)

here we know that the width of CI is 20

so

mean + z*sd/sqrt(n) - mean + z*sd/sqrt(n) = 20

2*2.58*262.2/20 = sqrt(n)

68 = sqrt(n)

n = 68*68 = 4624

c)

test stat = (xbar - mu) /(sd/sqrt(n))

(747.3 -800)/(262.2/sqrt(35)) = -1.18

now lets check the df = 35-1 = 34 , and alpha = 0.05

d) lets see the p value

The P-Value is .123

please note that this is a 1 tail test

as the p value is not less than 0.05 , hence we fail to reject the nul hypothesis

Please note that we can answer only 4 subaprts of a question at a time , as per the answering guidelines

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