Question 2: In men ages 65 and older, approximately 15% would test positive for

Question

Question 2: In men ages 65 and older, approximately 15% would test positive for prostate cancer if screened. Suppose a random sample of 5 men over the age of 65 is to be chosen, and let ! be the adjusted proportion that test positive for prostate cancer.

a) List all the possible values for ! and compute the probability of each value.

b) Suppose we randomly select 5 men over the age of 5, and 2 of them test positive for prostate cancer.

i. Compute the adjusted sample proportion " of men in this age range withprostate cancer.

ii. Construct a 95% confidence interval for the proportion of men in this age range who have prostate cancer.

iii. Using a significance level of , = 0.10, perform a Chi-squared goodness of fit test to test the hypothesis that 50% of men is this age group are positive for prostate cancer.

Explanation / Answer

Solution:

a. The possible values for ! are 0, 1, 2, 3, 4, 5. The probability is calculated using the binomial distribution formula are:-

P (X = x) =5Cx (0.15)^5 (1 - 0.15)^5 - x

P (X = 0) = 5C0 (0.15)^0 (0.85)^5-0 = 0.4437

P (X = 1) = 5C1 (0.15)^1 (0.85)^5-1 = 0.3915

P (X = 2) = 5C2 (0.15)^2 (0.85)^5-2 = 0.1382

P (X = 3) = 5C3 (0.15)^3 (0.85)^5-3 = 0.0244

P (X = 4) = 5C4 (0.15)^3 (0.85)^5-3 = 0.0022

P (X = 5) = 5C4 (0.15)^3 (0.85)^5-3 = 0.0000

b.

i.Adjusted sample proportion, " = X/n = 2/5 = 0.4

ii. 95% confidence interval is given by:-

p ± Z* p (1 – p)/n

0.4 ± 1.96*0.4*0.6/5

0.4 ± 1.96*0.2191

0.4 ± 0.4294

-0.0294, 0.8294

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