Suppose a new drug is available for treating high blood pressure. We want to investi its effect by running a controlled experiment: • We randomly select some patients with high blood pressure and randomly divide t into two groups, Group A and Group B.
Suppose a new drug is available for treating high blood pressure. We want to investi its effect by running a controlled experiment: • We randomly select some patients with high blood pressure and randomly divide t into two groups, Group A and Group B.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
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Transcribed Image Text:Suppose a new drug is available for treating high blood pressure. We want to investigate
its effect by running a controlled experiment:
. We randomly select some patients with high blood pressure and randomly divide them
into two groups, Group A and Group B.
1
● We then give the actual drug to patients in Group A and a placebo (i.e. non-harmful
fake drug) to patients in Group B.
Let X be the amount of blood pressure reduction in a patient taking the new drug, and
Y the amount of blood pressure reduction in a patient taking the placebo. Assume that
X Nux, 12) and Y~ N(μy, 8). How many patients (in total) do we need to select
for participation in this experiment, if we want to construct a 95% confidence interval for
"xy with length < 2?
![Let the following numbers represent the order statistics y₁, 92,927 of the n = 27
observations obtained in a random sample from a certain population of incomes (measured
in hundreds of dollars):
261 269 271 274 279 280 283 284 286 287 292 293 296 300
304 305 313 321 322 329 341 343 356 364 391 417 476
(i) Let Fx be the cumulative distribution function (CDF) of this population. Give a 95%
confidence interval for p = Fx (350) based on the given sample.
(ii) Give a point estimate for the 80th percentile of this population, To.8, based on the
given sample.
(iii) Suppose we use (yi, yj) as an interval estimate for 70.8. Express the approximate
confidence level for this interval in the form
P(70.8 € (Y₁, Yj)) ≈ P(N(0, 1) = [a, b])
where a, b are expressions involving i, j. To achieve a given confidence level 1 - a, we
may set a = -a/2, b = a/2 in the equation above. Solve for i and j in terms of Za/2.
(iv) Follow this procedure to give an approximate 95% confidence interval for 0.8 based
on the given sample. (Round i and j to nearest integer.)
(v) Compute the exact confidence level of the interval you found in the previous part.
(You probably want to use a computer for this.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2Fa8bc3194-67cc-4df4-854f-b0edf494cac9%2F06tcnj_processed.png&w=3840&q=75)
Transcribed Image Text:Let the following numbers represent the order statistics y₁, 92,927 of the n = 27
observations obtained in a random sample from a certain population of incomes (measured
in hundreds of dollars):
261 269 271 274 279 280 283 284 286 287 292 293 296 300
304 305 313 321 322 329 341 343 356 364 391 417 476
(i) Let Fx be the cumulative distribution function (CDF) of this population. Give a 95%
confidence interval for p = Fx (350) based on the given sample.
(ii) Give a point estimate for the 80th percentile of this population, To.8, based on the
given sample.
(iii) Suppose we use (yi, yj) as an interval estimate for 70.8. Express the approximate
confidence level for this interval in the form
P(70.8 € (Y₁, Yj)) ≈ P(N(0, 1) = [a, b])
where a, b are expressions involving i, j. To achieve a given confidence level 1 - a, we
may set a = -a/2, b = a/2 in the equation above. Solve for i and j in terms of Za/2.
(iv) Follow this procedure to give an approximate 95% confidence interval for 0.8 based
on the given sample. (Round i and j to nearest integer.)
(v) Compute the exact confidence level of the interval you found in the previous part.
(You probably want to use a computer for this.)
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