[Last update 7/21/04]
(7.1) REVIEW QUESTIONS
(A) Provide a synonym for paired samples.
(B) Besides "before/after" sampling, describe a way to derive a paired
sample.
(C) What is the "opposite" of a paired sample?
(D) How do independent samples differ from paired samples?
(E) What does EDA stand for?
(F) This chapter uses DELTA to represent the _____________ with paired measurements.
(G) True or false?: Suppose DELTA = X - Y, where
X = measurements after an
intervention and Y = measurements before an intervention. In this instance,
positive values represent increases after the intervention.
(H) What graphical techniques can be used to explore distributions of changes?
(I) What symbol is used to denote the mean difference parameter?
(J) What symbol is used to denote the mean difference estimator?
(K) How many degrees of freedom does the t confidence coefficient have when
studying 18 pairs?
(L) What t percentile is used with computing a 99% confidence interval?
(M) True or false: The confidence interval seeks to capture the sample mean.
(N) [M/C] A 95% confidence for that �d is (92.5, 98.5). Which is the better interpretation of this
result?
[(a) There is a 95% chance
that �d falls between 92.5 and 98.5 (b) There is a 95% chance that the interval captures �d.]
(O) Write the "general formula" for a confidence interval based on an
estimate that is approximately normally distributed.
(P) Write the null hypothesis for testing a paired difference for statistical
significance.
(Q) In assess n pairs, how many degrees of freedom does the t statistic have?
(R) Describe how to convert t statistics to a p value.
(S) The power of a test is its probability of avoiding a type ____ error.
(T) What is the value of F(0)?
(U) Is F(0.75) greater than 50% of less than
50%?
(V) Identify the meaning of the following symbols used in formula 7.3: D,
F, and s.
(W) What level of power is considered adequate, by convention?
(7.2) SEDDIQ (White blood cell counts determined by two
methods): A NASA experiment was set up to compare white blood cell count determinations made by two different
methods. Method 1 is a
CELDYNE method and method 2 is a UNOPETT method. Data representing WBC (� 1000/dl) are shown below.
(A) Explain why these data represent paired samples (as opposed to independent
samples).
(B) Calculate DELTA values for each
observation and then show these differences in the form of a stem-and-leaf plot. Describe the shape, location, and spread of the
distribution. Based on this graph, what do you conclude about the average results
of the two methods? What do you conclude about the consistency of
results?
ID CELDYNE UNOPETT
--- ------- -------
1 8.2 8.6
2 9.7 11.0
3 5.6 8.1
4 14.0 15.7
5 5.7 6.3
6 10.8 9.1
7 10.5 11.3
8 7.9 9.3
9 12.7 11.0
10 3.6 2.6
11 10.4 9.6
12 13.6 10.3
13 11.3 10.6
14 10.3 7.8
15 8.3 8.4
16 6.3 6.5
17 23.9 27.6
18 16.0 7.5
19 10.4 9.9
20 9.5 8.4
21 13.8 13.8
22 13.0 20.4
23 12.7 10.4
24 14.1 13.0
25 12.9 14.5
26 7.4 6.8
27 9.1 10.5
28 9.5 7.4
29 14.4 15.9
30 8.8 9.3
31 13.1 18.0
32 10.3 9.5
33 9.4 11.3
34 9.8 9.5
35 11.5 9.3
36 11.8 10.4
37 12.5 9.6
38 6.1 6.0
39 10.4 9.0
40 10.9 12.3
41 7.8 7.8
42 11.4 8.8
(7.3) OC-BP (Effect of oral contraceptive use on blood
pressure): Ten (10) women of childbearing age not currently using oral contraceptives have their systolic blood pressure measured. After a
year on oral contraceptives, the mean difference in blood pressure were 5.2 mm Hg higher
while on oral contraceptives (sd = 4.88).
(A) Calculate a 95% confidence interval for the mean change in systolic blood
pressures. Interpret this result.
(B) Test to see whether the mean change in blood pressures is
significant at a = .05. Show all hypothesis testing
steps. (Suggestion: To help determine the approximate p value, draw the t
distribution and place it at an appropriate location on the curve. Then shade the
regions corresponding to the p value and place one or two appropriate t
percentile landmarks on the curve to help ascertain the area of the p value
regions.)
(7.4) FERTILITY (Fertility and oral contraceptive use):
A topic of recent interest in women's health is the effect of different contraceptive methods on fertility.
Suppose we want to
compare how long it takes users of oral contraceptive to become pregnant after stopping
contraception with that of cervical diaphragm users. To study this
question, 20 oral contraceptive users and 20 diaphragm users are matched on age, race, parity, and socioeconomic status. The mean difference in time to fertility
in diaphragm users compared with OC users is 5 months
with sd = 8 months.
(A) Calculate a 95% confidence interval for �d. Interpret
this interval.
(B) Determine whether the observed mean difference is significantly at a =
.05. As always, show all hypothesis testing steps. (Include a drawing of the t
distribution, as specified in the prior problem.)
(7.5) FLUORIDE (Cavity rates before and after city water
fluoridation): Data below represent the number of cavity-free children per 100
BEFORE and AFTER city water fluoridation
projects (Osborne, 1980, p. 40). The mean cavity-free rate before fluoridation was 26.13 per 100. After fluoridation, the mean
rate was
38.34. Thus, the increase was 12.21 (sd = 13.62).
(A) Calculate differences (DELTAs) in
cavity-free rates. Then, plot this information as a stem-and-leaf plot. Interpret your
plot by addressing its shape, central location and spread (range).
(B) What
percentage of cities failed to show an improvement?
(C) Conduct a test to determine whether the change in cavity-free rates is
significant at a = .05. Show all hypothesis testing steps.
CITY BEFORE AFTER
---- ------- -----
1 18.2 49.2
2 21.9 30.0
3 5.2 16.0
4 20.4 47.8
5 2.8 3.4
6 21.0 16.8
7 11.3 10.7
8 6.1 5.7
9 25.0 23.0
10 13.0 17.0
11 76.0 79.0
12 59.0 66.0
13 25.6 46.8
14 50.4 84.9
15 41.2 65.2
16 21.0 52.0
(7.6) COTININE2 (Decreases in salivary cotinine levels over time): Cotinine, a by-product of tobacco, is found in the saliva of smokers for a variable time following tobacco exposure. As a study of tobacco detection, salivary cotinine levels were measured in volunteers 12- and 24-hours following exposure. Data are:
ID 12 hours 24 hours
--- -------- ---------
1 83 34
2 68 27
3 68 29
4 98 29
5 30 14
6 24 9
7 41 11
(A) Plot the changes (DELTA)
in cotinine levels. Discuss the central location, spread, and shape of
the distribution.
(B) Estimate the mean change in cotinine levels with 95% confidence.
(7.7) BPH-SAMP (Quality of life following treatment of benign prostate hyperplasia): Benign prostate hyperplasia (BPH) is a non-cancerous enlargement of the prostate gland affecting the quality of life for millions of men worldwide. The condition causes restriction of the flow of urine from the bladder and adversely affects the quality of life of those effected. A study of a minimally invasive procedure for the treatment of BPH looked at pretreatment quality of life at the time of treatment (QoLTX) and quality of life after 3 months of treatment (QoL3Mo). Patients were asked to rate their quality of life as follows: 0 = Delighted, 1 = Pleased, 2 = Mostly Satisfied, 3 = Mixed, 4 = Mostly Dissatisfied, 5 = Unhappy, 6 = Terrible. Data for 10 randomly-selected patients participating in the study are :
ID QoLTX QoL3Mo
-- ----- ------
1 2 1
2 4 1
3 3 1
4 4 3
5 5 2
6 6 2
7 4 2
8 4 5
9 3 3
10 3 1
(A) Why should we consider these paired samples (as opposed to independent
samples)?
(B) Calculate changes (DELTA values) in
quality of life over the 3 months of study.
(C) Calculate the mean, sum of squares, standard deviation and standard error of
the mean for the DELTA values.
(D) Determine whether the change is significant at a =
.05.
(7.8) FAUXPAS (Social skills in high schoolers): At the beginning a program designed to promote social skills in high school students, we take 8 high-schoolers to a shopping mall and count the number of socially inappropriate behaviors (e.g., put-downs and other unlovable behaviors that each of us engaged in when we were that age) exhibited by each student. After two weeks in a program during which we attempt to instill in our subjects an array of social behaviors that we think they should adopt while omitting social behaviors we don't wish to nurture, we take the same group to the mall and again count the number of inappropriate behaviors per student. Data are:
ID VISIT1 VISIT2 DELTA
-- ------ ------ ------
1 5 4
-1
2 13 11 -2
3 17 12 -5
4 3 3
0
5 20 14 -6
6 18 14 -4
7 8 10
2
8 15 9
-6
(A) Plot the DELTA variable in the form of a stem-and-leaf plot.
Describe the shape, central
location, and spread of the distribution.
(B) What percentage of students failed to show an improvement?
(C) Calculate the mean change in faux pas, and its standard deviation.
Interpret these results.
(D) Test the data to see if there has been a significant change in the number of socially inappropriate behaviors. Let a = .05.
(7.9) VIT-C (Vitamin C and the Common Cold): An investigator wishes to determine whether vitamin C reduces the frequency of the common cold. To reduce the effects of environmental variability, the investigator pairs children from the same family and randomly assigns to one of the pair members high doses of vitamin C. The other pair member gets a placebo. The investigator keeps track of the number of day in which upper respiratory symptoms were observed over the year. Data, representing the number of sick days, are shown below. Explore the observed difference in cold frequency and test this difference for significance.
Pair Vit C Placebo
----- ----- ------
1 2 3
2 5 4
3 7 9
4 0 3
5 3 6
6 3 5
(7.10) BLANK
(7.11) DELTAPOW1: A study of 25 pairs fails to find a significant difference. The standard deviation of the difference is 10. Assuming a = .05 (two-sided), what was the power of the study to detect a mean difference of 1? How would this result influence your interpretation of the non-significant results?
(7.12) DELTAPOW2: Reconsider the problem presented in 7.11, but now determine the power of this study to find a difference of 10? How would you interpret this power calculation?
Key to Odd Numbered Problems
Key to Even Numbered Problems (may not be posted)