Question

A medical researcher would like to determine how effective a new form of knee surgery is for
treating knee pains. (In particular, is there a relationship between surgery status and pain
reduction status?) He gathers 500 people with knee pain who are scheduled to have this
surgery, and 500 people with knee pain who are not scheduled to have this surgery. He follows
up with both groups after 2 months and determines the number of participants of each group
who report a “significant reduction in knee pain”. He summarizes his findings in the table
below:
Contingency Table Pain Reduction Status
Pain Reduction No Pain Reduction
Surgery Status Surgery 340 160
No Surgery 280 220
(a) (2 points) Between a t-test, a χ
2
-test, and an F-test, which is the most appropriate for
this study?
(b) (1 point) State the hypotheses for this test.
(c) (1 point) What is/are the conditions(s) for this test to be valid?
(d) (1 point) Are the conditions for this test satisfied? (Yes / no)
(e) (3 points) The researcher finds a test statistic of 15.65 and a p-value of 0.0001. Interpret
this p-value. (Hint: 0.0001 is the probability of...)
(f) (1 point) Based on a significance level of 0.001, should we reject or fail to reject H0?
(g) (1 point) Is this an experiment or an observational study?
(h) (1 point) If this is an experiment, is this study single-blind, double-blind, or neither? If
this is not an experiment, leave this question blank.
(i) (1 point) If this is an experiment, is this a randomized controlled trial? If this is not an
experiment, leave this question blank

(j) (1 point) If this is an observational study, is it prospective or retrospective? If this is not
an observational study, leave this question blank.
(k) (1 point) If this is an observational study, is it case-control or cohort? If this is not an
observational study, leave this question blank.
(l) From prior experience, the medical researcher thinks that about 35% of new surgeries he
tests are effective in pain reduction. Suppose this researcher conducts 100,000 of these
tests (for all different surgeries), at a significance level of 0.001 and a power of 0.55.
i. (1 point) When H0 is true, what is the probability of rejecting H0 for these tests?
ii. (1 point) When H0 is false, what is the probability of rejecting H0 for these tests?
iii. (2 points) For this test, what is a Type I error, and what is a Type II error?
iv. (2 points (bonus)) What is the overall (average) error rate?
v. (1 point (bonus)) Amongst all rejections of H0, what is the (average) error rate?
vi. (1 point (bonus)) Amongst all failures to reject H0, what is the (average) error rate?
vii. (1 point (bonus)) In these 100,000 tests, how many Type I errors (on average) are
made?

(m) The researcher also wants to estimate the proportion of surgery recipients who report pain
reduction after two months. In the sample of 500, 340 report pain reduction after two
months, so ˆp = 64%.
i. (1 point) What is the population of interest for this confidence interval?
ii. (1 point) It is thought that between 200,000 and 300,000 people will receive this
surgery. Is a finite population correction appropriate here?
iii. (2 points) The researcher calculates a 2.14% margin of error for his desired 95% con-
fidence interval, without a finite population correction. Use this to construct the 95%
confidence interval, including the FPC if appropriate.
iv. (1 point) The researcher claims that this interval has a 95% probability of containing
p. Is this correct?
v. (1 point) The researcher’s friend says that the researcher was mistaken, and that the
interval in fact only has a 95% probability of containing ˆp. Is his friend correct?
vi. (2 points) Suppose the researcher would like to have a margin of error of only 1%.
What is the minimum sample size required to obtain this result?
vii. (2 points) Confidence intervals for p require the distribution of ˆp to be (approximately) normal. What do we call the distribution of a sample statistic, such as ˆp?
What do we call the standard deviation of the distribution of a sample statistic?

(n) (2 points) The researcher is deciding between three different sampling techniques. Rank
these sampling techniques from least biased to most biased.
A. The researcher obtains a list of all patients scheduled to have this surgery in the next
12 months. He then picks 500 names from this list randomly.
B. The researcher travels to the most expensive hospital in the area. He then picks the
next 500 patients scheduled to receive this surgery.
C. The researcher puts up fliers around local malls, shopping centres, grocery stores, gas
stations, hospitals, and universities, advertising his study. He then picks the first 500
respondents to his fliers that qualify for the study

Answer 1

A. Chi square test is most appropriate.

B. H₀: Status of knee pain and surgery are independent

C. A key assumption of the chi square test of independence is that each subject contributes data to only one cell. Therefore the sum of all cell frequencies in the table must be the same as the number of subjects in the experiment.

D. This assumption is satisfied in this situation.

E. 0.0001 is the probability of observing a test statistic as extreme as 15.65, assuming the null hypothesis is true.

F. Based on significance level 0.001, we reject H₀. Because p value is less than significance level.

G. This is an experiment.

H. This is neither single blind nor double blind.