Exercises

(1) ELECT: Political Survey

A pre-election survey of 100 prospective voters shows 55 in favor of Candidate A.

(A) Calculate p^.
(B) Assuming data represent a simple random sample of the electorate, calculate a 95% confidence interval for the proportion of all voters who are in favor of candidate A.
(C) Test the hypothesis that at least 50% of the potential electorate will vote for candidate A. (State H0 and H1 assuming a two-sided alternative; set alpha; report your hypothesis testing statistics, whether it be an exact binomial probability or confidence interval; state your conclusion.)
(D) Woud a declaration of victory on the part of Candidate A be premature? Explain.

(2) BREASTCA: Breast Cancer in Offspring of Cases

We know that approximately 2% of women develop breast cancer between the ages of 50- and 54. Suppose we find 32 incident cases of breast cancer in a sample of 1000 women between the ages of 50- to 54. These women are the daughters of women who had breast cancer.

(A) Calculate the incidence of breast cancer in the sample.
(B) Calculate a 95% confidence interval for the incidence of breast cancer in 50- to 54-year old daughters of breast cancer cases.
(C) Test whether this incidence is significantly greater than expected? (List H0 and H1, set alpha; compute a test statistics; state your conclusion.)
(D) Interpret your results.

(3) TOUCH: Therapeutic Touch

An study by an 11-year old girl made headlines for challenging the validity of therapeutic touch (Rosa et al., 1998). In therapeutic touch, the touch therapists' hands are passed over a patient's body without actually touching the patient, supposedly to manipulate human energy fields. In the current experiment, touch therapists rested their hands, palm up, on a flat survace approximately 25 to 30 cm apart. To prevent the experimenter's hands from being seen, a tall, opaque screen with cutouts at its base were placed over each subject's arm, and a cloth towel was attached to the screen and draped over the therapists' arms. Each therapist underwent a test of 10 trials in which the experimenter (the 11-year-old girl) hovered her right hand, palm down, 8 to 10 cm above one hand of the therapist and then said "Okay." The touch therapist then stated which of his or her hands was nearer to the experimenter's hand. Each subject was permitted to take as much or as little time as necessary to make each determination. Results are as follows:
No. correct (out of 10) Frequency Frequency * No. Correct
0 0 0
1 1 1
2 1 2
3 8 24
4 5 20
5 7 35
6 2 12
7 3 21
8 1 8
9 0 0
10 0 0
Total 28 123

Notice that this study found that touch healers were able to detect the presence of the experimenter in 123 out of 280 trials. However, let us view each of the above 28 trials as a cluster (e.g., one cluster showed a results of 1 of 10, one cluster showed a results of 2 of 10, and so on). We want to calculate a 95% confidence interval for the proportion of "successes." To do this, we must calculate the design effect (deff) attributable to the clustered sample.

(A) Calculate the point estimate for p (the probability of success).
(B) Calculate a confidence interval for p, while ignoring the clustered design (tentatively assuming that data were derived independently).
(C) The design effect attributable to clustering is computed below (deff = 1.12) Now use EpiTable to calculate the proper 95% confidence interval taking into account the effect of clustering.
(D) Is there any evidence to contradict the hypothesis of random selection ("detection") of a human energy field? Discuss the study's results.

  Describe  Compare  Study  Sample  Probability  Setup
-[_]------------------------------------------------------
 Clust Num  Den      Num  Den      Num  Den      Num  Den
 No. 1  1    10   13  4    10   25  7    10   37  0    0
 No. 2  2    10   14  4    10   26  7    10   38  0    0
 No. 3  3    10   15  4    10   27  7    10   39  0    4
 No. 4  3    10   16  5    10   28  8    10   40  0    0
 No. 5  3    10   17  5    10   29  0    0    41  0    0
 No. 6  3    10   18  5    10   30  0    0    42  0    0
 No. 7  3    10   19  5    10   31  0    0    43  0    0
 No. 8  3    10   20  5    10   32  0    0    44  0    0
 No. 9  3    10   21  5    10   33  0    0    45  0    0
 No.10  3    10   22  5    10   34  0    0    46  0    0
 No.11  4    10   23  6    10   35  0    0    47  0    0
 No.12  4    10   24  6    10   36  0    0    48  0    0
 

 Global variance               :  0.000880
 Cluster variance              :  0.000988
 Design effect                 :      1.12

(4) PREGRATS: Malformation Rates in Rats

In a laboratory investigation of the teratogenicity of an agent, we find malformed pups in 12 out of 85 lab rat litters.

(A) Calculate p^.
(B) Report a 95% confidence interval for p.
(C) In rats we normally expect a malformation rate of about 5%. Do the current data provide significant evidence of teratogenicity? (Perform a formal hypothesis test, listing all steps in the process.) Interpret your results.

(5) SMOKE.REC: Smoking Recidivism Rates (Source: Rosner, 1990, p. 104)

Smoking cessation programs have modest success in helping their clients stop smoking. A study of 234 smokers who had expressed a desire to stop smoking recorded the number of days each client successfully restrained from smoking (variable DAYS).

(A) Convert this outcome to a binary variable (IF DAYS < 365 THEN RECID = 1 ELSE RECID = 2) and then calculate the recidivism "rate" in the sample.
(B) Report a 95% confidence interval for this rate.
(C) Historical norms suggest that approximately 90% of smokers return to smoking after having expressed a desire to quit. Do the current data provide significant evidence of this program's superiority? (Conduct a formal hypothesis test, including all steps of the analysis from statements of the null and alternative to the test's conclusion.)
(D) Interpret the above analyses as a whole.

(6) BINSIZE: Binomial Sample Size Exercise

(A) Determine the sample size required to calculate a 95% confidence interval for p with a margin or error (d) of no more than 10% assuming a population of one million and expected proportion of 50%.
(B) Recalculate the sample size requirements of the above study letting d equal 5%.
(C) Same as above, except let d equal 3%.
(D) Same as above, except let d equal 2%.
(E) Draw a sample size curve using the above calculations. Let the x-axis represent margin of error d and the y-axis represent the required sample size n.
(F) Repeat the above calculations, now assuming p = 10%.
(G) Draw the sample size curve on the same axes used previously, and describe how this lower assumed proportion affects your sample size estimates.

(7) DBP: Diastolic Blood Pressure (Source: Osborn, 1979, p. 22, modified)

Data, below, represent diastolic blood pressure measurements in a sample of 50 men. Let diastolic blood pressure 95 mm Hg represent hypertension. Calculate the prevalence of hypertension in the sample and a 95% confidence interval for the prevalence of hypertension in the population from which this sample was drawn.

94  94  82  100  112 110   84  78  92  112 
94  92  86   84   90  72   88  92  88   84
 
98  98  84   90   90  70   80  90  80  106 
74  95 100   94   84  70  102  92  84   80
84  86  98   82   80  88   80  84 100   86



(8) EDENTITION: Edentition in England (Source: Osborn, 1979, p. 24, modified)

In a report of adult dental health in 25- to 34-year-old English women showed 20 of 262 women with edentition (missing teeth).

(A) Estimate the edentition rate in England and Wales and calculate a 95% confidence interval for this rate.
(B) In the U.S., we expect 6.1% of women in this age range to show edentition. Test whether the English rate differs significantly from the U. S. rates. (Perform a complete hypothesis test, stating the null and alternative hypothesis, set alpha, report an appropriate hypothesis testing statistic, state your conclusion.)

(9) ABUSE: Childhood Abuse of Psychiatric Patients (Source: Daniel, 1995, p. 174, modified)

In a study of physical and sexual abuse during childhood, Brown and Anderson found 166 people with histories of abuse in a sample of 947 psychiatric patients.

(A) Compute the childhood abuse rate in the sample. Include a 95% confidence interval for p.
(B) Assume that the prevalence of physical and sexual abuse during childhood in the general population is 3%. Based on this assumption, can we say that psychiatric patients suffer a significantly higher rate of abuse than the population as a whole?

(10) THINK!: What Additional Information Is Needed?

Suppose we observe 3 events (cancers, side-effects, bad outcomes, "whatever") in a population, and would like to know whether this occurrence is significant. What additional information would you need to make a judgement under such circumstances?

(11) PUZZLER: This Week's Puzzler

Suppose you are suspicious that a coin is "unfair" (i.e., that it doesn't turn up an equal number of heads and tails upon flipping). In particular, you are suspicious that the coin turns up heads more than it should. To test this problem, what would your null and alternative hypotheses be? Would you use a one-sided or two-sided alternative? (This, in a way, will say a lot about your philosophy.) How many times would you flip the coin? Carry out an experiment to test a coin.