Question

1. It is recognized that cigarette smoking has a deleterious effect on lung function. In a study of the effect of cigarette smoking on the carbon monoxide diffusing capacity (DL) of the lung, researchers found that current smokers had DL readings lower than either ex-smokers or non-smokers. The DL readings for a random sample of n = 20 current smokers yielded ¯ x (the bar is supposed to be over the x) = 89.87 and s = 14.91.
(a) The average DL reading for non-smoker is known to be 100. Set up H0 and Ha to formally test whether the DL readings of current smokers are lower than non-smokers.

H0>=100

Ha<100

(b) Use α = .05 and suppose it is known that the DL readings are normally distributed and σ = 15. Specify the test statistic and the decision rule. Specify the rejection region in terms of ¯ X. Note that you do not need the data at this point.

z=((¯ x-100)/(15/sqrt(20))) < -1/645

(c) Plug the data into your decision rule, do you accept or reject H0? What is the p-value?

(89.87-100)/(15/sqrt(20))=-3.020

Reject the null hypothesis

p=0.005

2. Refer to Problem 1 above with σ = 15 known.
(a) Suppose the average DL reading for current smokers is actually µ1 = 90. Calculate the β-risk (the probability of committing a type-II error) for the decision rule of 1(b).

β-risk =0.0901

(b) Repeat the above calculation for µ1 = 97.

β-risk =0.7734

3. Refer to Problem 1 above with σ unknown.
(a) Specify the test statistic and the decision rule.

(b) Plug the data into your decision rule and make a decision. What is the p-value?

4. Refer to Problem 1 above.
(a) Calculate a CI for µ the average DL reading for current smokers.

89.87+-6.973

(82.897, 96.843)

(b) Suppose one wishes to construct a 95% CI no wider than 5 (total length), how large a sample should be drawn? You may use σ = 15 as a rough estimate for the purpose.

n>=97.4

*** I believe I have figured out question 1, 2 and 4, but I need help with the 3. Any direction is appreciated. ***

Answer 1

3.

(a)

Decision rule : Reject Ho if Test statistic(t) < - 1.729

(b)

Since, Test statistic , t = - 3.038 < - 1.729 , We Reject the null hypothesis.

P-value = 0.00338

4.

(a)

95 % CI for the average DL reading for current smokers when .

95 % CI for the average DL reading for current smokers when .

(b)

Now, we take margin of error < 5

We have,

We need to solve for n.

So, If one wishes to construct a 95% CI no wider than 5 (total length), a sample size of n = 35 ( approx) should be drawn.