Download STA220H1 L0301 Practice Problems – November 11

STA220H1 L0301 Practice Problems – November 11-13, 2014
1. (a) If the null hypothesis is true, how often should the P-value be less than 0.05?
(b) If the power of a test for the mean is 0.979 for a particular alternative value of the mean, what is
the Type II error rate for this mean?
(c) Intuitively, why are we more likely to reject H0 when the true mean is further from the null value?
(d) If the null hypothesis is false, which one of the types of errors (Type I or Type II) can you not
make?
(e) The test statistic for a two-sided test of a proportion is 2.5. Will the P-value be larger if: (1)
n = 50 or (2) n = 100 (for the same value of the test statistic)?
(f) In a two-sided test for a proportion, how do we get larger power with larger sample size (for the
same hypothesis, and same sample proportion)?
(g) The test statistic for a two-sided test of a mean is 2.5. Will the P-value be larger if: (1) n = 50
or (2) n = 100 (for the same value of the test statistic)?
(h) Will the probability of making a Type II error when the alternative hypothesis is true be larger
when the significance level is: (1) 0.05 or (2) 0.10?
(i) True or False: Suppose the null hypothesis is µ = 5 and we fail to reject H0 . Under this
scenario, the true population mean is 5.
(j) True or False: With large sample sizes, even small differences between the null value and the
true value of the parameter will be identified as statistically significant.
(k) True or False: You are more likely to make a Type II error when using a small sample size
than when using a large sample size.
(l) Imagine a situation where you are carrying out a statistical test. Keeping everything else the
same, who does the power change if you carry out a one-sided versus a two-sided test?
2. (From OpenIntro Exercise 4.29)
Suppose drug regulators monitored 403 drugs last year, each for a particular adverse response. For
each drug they conducted a single hypothesis test with a significance level of 5% to determine if the
adverse response rate was higher in those taking the drug than those who did not take the drug. The
regulators ultimately rejected the null hypothesis for 42 drugs.
(a) In the case of a drug where the null hypothesis was rejected, what type of error might the regulators
have made? Describe what it means in practical terms.
(b) In the case of a drug where the null hypothesis was not rejected, what type of error might the
regulators have made? Describe what it means in practical terms.
(c) Suppose that the vast majority of the 403 drugs do not have adverse effects. Then, if you picked
one of the 42 suspect drugs at random, about how sure would you be that the drug really has an
adverse effect?
(d) Can you also say how sure you are that a particular drug from the 361 where the null hypothesis
was not rejected does not have the corresponding adverse response?
3. (From OpenIntro Exercise 4.27)
A patient named Diana was diagnosed with Fibromyalgia, a long-term syndrome of body pain, and
was prescribed anti-depressants. Being the skeptic that she is, Diana didn’t initially believe that antidepressants would help her symptoms. However after a couple of months of being on the medication,
she decides that the anti-depressants are working, because she fells like her symptoms are in fact getting
better.
(a) Write the hypotheses in words for Diana’s skeptical position when she started taking the antidepressants.
(b) What is a Type I error in this context?
(c) What is a Type II error in this context?
(d) How would these errors affect the patient?
4. Wait times in a hospital ER are being analyzed. The previous year’s average was 128 minutes. We
want to know if the mean wait time has increased (so, unlike the example in class, proceed with a
one-sided test).
(a) What are the null and alternative hypotheses?
(b) If we plan to collect a sample size of n = 64, what values could x be so that we will reject H0 ?
Suppose that the standard deviation is 38 minutes. You can assume that the sampling distribution
of x is very close to Normal.
(c) Suppose that this year’s average wait time is 137.5 minutes (for all patients). Calculate the
probability of a Type II error for our statistical test, which is based on a sample of 64 patients.
(d) Calculate the power of the test at detecting an increase of 9.5 minutes in the average wait time.
(e) Suppose you wanted more power? What could you do?
(f) Suppose now that this year’s average wait time is 129 minutes. How will this affect the power?
(g) Suppose we decide to collect data on more patients, so that we have a sufficient sample size to
have 90% power for the test that mean wait time has increased, when in fact it has increased by
1 minute. What concerns might you have about this approach?
5. From an old STA220 exam:
We want to test H0 : µ = 5 vs HA : µ 6= 5 at α = 0.05 with a t test based on a simple random sample
of n = 4 observations. For purposes of estimation, we are willing to guess that σ = 1.0. Let β = the
probability of a Type II error = 0.3 when µ = 4. Which of the following statements are true?
I. The power of this study is 0.7 for detecting a population mean equal to 4.
II. β > 0.3 when µ = 3.
III. If we increase the sample size to 15 then β < 0.3.
IV. β > 0.3 when µ = 4 and n = 4 if σ were twice as large (that is, σ = 2.0).
A. All
B. All but I
C. All but II
D. All but III
E. All but IV