Midterm II Solutions -
Introduction to Statistics
-
Spring 2016
1. Suppose that 60% of students in a class are women (the rest are men). Also suppose that 20% of
the students are gay. If 70% are either female or gay, what percentage of the class are gay females?
Solution: P(F or G) = P(F) + P(G) – P(F & G)
0.7 = 0.6 + 0.2 – P (F & G) = 0.8 – P (F & G)
P(F & G) = 0.8 – 0.7 = 0.1 → 10% are female and gay
2. If
,
, and
, then
?
Solution:
3. Given the table, calculate the expected value.
Policyholder
Outcome
Payout
x
Probability
P(X=x)
Death
600
0.002
Disability
100
0.02
Solution:
4. Calculate the standard variation of the payout of problem 3.
Solution:
5. If
and
, then
Solution:
6.
Suppose we find that the proportion of right-wing authoritarians in a group of 2000 people is 0.35.
Find the standard deviation of the proportion of right-wing authoritarians that would be expected in
a sample of this size?
Solution:
7. If a random variable X has standard deviation of exactly 200, then what is the standard deviation of
a sample of size 40,000 drawn from X?
Solution:
8. Suppose we construct a confidence interval for means (from a sample of size 12,345), based on a
critical value of 0.025. What percent of the time will the true population mean be in our interval?
Solution: 95%
9.* A pilot study showed that 0.3% of credit card offers in the mail end up with a person signing up.
To be within 0.1% of the true rate with 95% confidence, how big does the test mailing have to
be? (Use 1.96 for your the z-value needed for the 95% confidence interval.)
Solution:
n has to be at least 11491
10. A widget factory had a failure rate of 30% (cracked widgets).
Find the standard deviation of this model, for a sample size of 500.
Solution:
11. Suppose they obtained a new machine that supposedly had a better success rate. In random
sample of 200 widgets, 50 widgets were found to be cracked. What's the z-score?
Solution:
12. If the P value for the z score you just found was 0.007348715 (the probability of seeing a z score
that low or lower), should you reject the null hypothesis at the 95% confidence level?
Solution: Yes, since 0.0073487 < 0.05. That new machine does appear to do a better job!
13. Which of the following defines the concept of P-value?
(a) P(observed stat value |
is true)
(b) P(
is true | observed stat value)
(c) P(observed stat value |
is true) ← Answer
(d) P( is true | observed stat value)
(e) None of these
14. True or False:
An example of a Type 1 error is a patient diagnosed as having a disease that she doesn't really have.
Answer: True
15. For a study that hopes to determine if exercise for 40 minutes per day changes the incidence of
panic attacks in group of anxiety patients (which has been a mean of 50 panic attacks per week), please
write out a null hypothesis and alternative hypotheses? Let p be the number of panic attacks per
week.
Answer:
:
:
16.* Suppose that 500 out of 800 women prefer ovals to rectangles, while only 300 out of 900 men
prefer ovals. Find the standard error of the difference of proportions between these two groups.
Solution: Let
, so
, and
, so
,
and
.
Then,
17. Use
, for a 95% confidence interval, and compute a 95% confidence interval for the
difference in proportions featured in problem 16. Round off to the nearest thousandth.
Solution:
95% confidence interval =
18. If
:
, should we reject
Solution: We must reject
on the basis of your results from problem 18? Why?
, since 0 is not within the 95% confidence interval.
19. Which data are paired data suitable for paired t-test, and which are suitable for a 2-sample t-test?
Diet additive effect on hen egg-laying rates.
Proportion of fundamentalists in two groups of people.
20. For data suitable for paired t-test, suppose that
,
Answer: paired t-test
Answer: 2-sample t-test
, and
. Then SE=?
Solution:
21. Continuing with problem 20, suppose you find that
, corresponding to 399 degrees of
freedom and a 99% confidence interval. Construct the confidence interval.
Solution:
95% confidence interval
22. If
:
, should we reject
on the basis of your results from problem 21? Why?
Solution: We reject , since 0 is not within the 99% confidence interval,
and in fact is far from it!