deaths of infants in the US in the first year of life

Part 1.deaths of infants in the US in the first year of life (9 points) In the US, the probability that a child dies in the first year of life is 0.0085.
In a certain rural county, for a recent year , 950 infants were born.
(For the sake of this exam, assume that these are all singleton births.)
18 of these infants died in their first year of life.
The county supervisors ask you if this indicates that the county’s experience is different from
that of the US population, if too many children are dying in their first year of life.
1. (4 points) Explain what probability distribution you will use to assess these data, and exactly
why. Discuss all relevant features and assumptions. 2. (3 points) If the probability of dying in the first year of life for infants born in this county is the
same as the US probability, what is the probability that 18 or more children of the 950 born
would die in their first year of life? Give all details. 3. (2 points) Based on your probability calculation, what you do conclude?
Write a summary sentence for the county supervisors. Part 2.reasoning about disease from a test result (14 points) Physicians often order a lung biopsy for individuals with chronic persistent cough and no
indication of disease on chest X-ray. However, routine lung biopsies cannot differentiate
between lung cancer and another, non-fatal lung disease called sarcoidosis. An abnormal
biopsy finding could be either disease and so requires follow-up testing.
Notation: abnormal biopsy A lung cancer C sarcoidosis S no lung disease N Suppose for men in their 60s with chronic persistent cough who have never smoked tobacco
we know the following: and P( A | C ) = .95 P( A | S ) = .90 P( A | N ) = .005 P( C ) = .005 P( S ) = .035 P( N ) = .96 1. (4 points) Set up a tree diagram to display this information, with the top set of branches
representing cancer, sarcoidosis, and no lung disease and the second set of branches the biopsy
outcome. 2. (6 points) Find the overall probability that a man in his 60s with chronic persistent cough who
has never smoked tobacco gets an abnormal biopsy. 3. (4 points) Find the probability of sarcoidosis (S) given the biopsy is abnormal (A). Part 3.power and sample sizes (15 points) common scenario for questions 1 to 5:
The investigators plan to use a sample of size 60
to carry out tests of hypotheses with the null hypothesis µ = 250 .
They think the population standard deviation is equal to 50.
They really want their hypothesis tests to correctly reject the null hypothesis if µ = 235 .
Stata use is suggested!
1. (2 points) If the investigators use ? = .05 and a one-sided alternative hypothesis µ<250,
what is the power of the test? Show how you found the answer. 2. (2 points) If the investigators use ? = .01 and a one-sided alternative hypothesis µ < 250,
what is the power of the test? Show how you found the answer. 3. (1 point) Compare your answers to questions 1 and 2 and comment on the difference. 4. (2 points) If the investigators use ? = .05 and a two-sided alternative hypothesis µ ? 250,
what is the power of the test? Show how you found the answer. 5. (1 point) Compare your answers to questions 1 and 4 and comment on the difference. 6. (3 points) Assume that the population mean really is 235. What size sample would be needed
to have 90% power to correctly reject the null hypothesis µ =250 if the investigators use a onesided alternative hypothesis µ < 250, using ? = .05? Continue to use 50 as an estimate of
the population standard deviation. Show how you found the answer. 7. (3 points) Assume that the population mean really is 235. What size sample would be needed
to have 90% power to correctly reject the null hypothesis µ = 250 if the investigators use a twosided alternative hypothesis µ ? 250, using ? = .05 ? Continue to use 50 as an estimate of
the population standard deviation. Show how you found the answer. 8. (1 point) Compare your answers to questions 6 and 7 and comment on the difference.