Multiple Choice Questions
Inference - Two independent samples on means
1. In a study of iron deficiency among infants, random samples of infants
following different feeding programs were compared. One group contained
breast-fed infants, while the children in another group were fed by a standard baby formula without any iron supplements. Here are summary
results of blood hemoglobin levels at 12 months of age.
Group
Breast-fed
Formula-fed
Sample Size
8
10
Sample Mean
13.3
12.4
Sample
1.7
1.8
Std. Deviation
A 98% confidence interval for the mean difference in hemoglobin level
between the two populations of infants is:
(a) 0.9 ± 1.94
(b) 0.9 ± 2.08
(c) 0.9 ± 2.13
(d) 0.9 ± 2.15
(e) 0.9 ± 1.63
Solution: d
Past performance 1989 Dec - 64% (14% a,c)
Past performance 1990 Dec - 73%
2. A study was conducted to investigate the effectiveness of a new drug for
treating Stage 4 AIDS patients. A group of AIDS patients was randomly
divided into two groups. One group received the new drug; the other
group received a placebo. The difference in mean subsequent survival
(those with drugs - those without drugs) was found to be 1.04 years and
a 95% confidence interval was found to be 1.04 ± 2.37 years. Based upon
this information:
(a) We can conclude that the drug was effective because those taking the
drug lived, on average, 1.04 years longer.
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(b) We can conclude that the drug was ineffective because those taking
the drug lived, on average, 1.04 years less.
(c) We can conclude that there is no evidence the drug was effective
becaue the 95% confidence interval covers zero.
(d) We can conclude that there is evidence the drug was effective because
the 95% confidence interval does not cover zero.
(e) We can make no conclusions because we do not know the sample size
nor the actual mean survival of each group.
Solution: c
Past performance 1990 Dec - 79%
Past performance 1998 Dec - 77%
Past performance 2006 Dec - 85%
3. Samples of hamburger were selected from two different outlets of a large
supermarket to measure the percentage of fat present in the meat, with
the following summary data.
n
mean
std.dev
Outlet 1
5
10.3
1.6
Outlet
10
10.7
2.3
2
percent
percent
It is reasonable to believe that both outlets have the same variability.
Hence, the pooled standard deviation is:
(a) 1.95
(b) 2.08
(c) 4.38
(d) 2.09
(e) 2.11
Solution: e
Past performance 1989 Dec - 72%
4. The degrees of freedom of the pooled estimate in the previous question is:
(a) 15
(b) 13
(c) 7.5
(d) 5
(e) 10
c
2006
Carl James Schwarz
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Solution: b
Past performance 1989 Dec - 90%
5. A study was conducted to estimate the effectiveness of doing assignments
in an introductory statistics course. Students in one section taught by
instructor A received no assignments. Students in another section taught
by instructor B, received assignments. The final grade of each student was
recorded. A 95% confidence interval for the difference in the mean grades
(Section A - Section B) was computed to be −3.5 ± 1.8. This means:
(a) There is evidence that doing assignments improves the average grade
because the difference in the population means is less than zero.
(b) There is little evidence that doing assignments improves the average
grade because the 95% confidence interval does not cover 0.
(c) There is evidence that doing assignments improves the average grade
because the 95% confidence interval does not cover 0.
(d) There is evidence that doing assignments does not improve the average grade because the 95% confidence interval does not cover 0.
(e) There is little evidence that doing assignments does not improve the
average grade because the 95% confidence interval does cover 0.
Solution: c
Past performance 1989 Dec - 73%
6. Popular wisdom is that eating pre-sweetened cereal tends to increase the
number of dental caries (cavities) in children. A sample of children was
(with parental consent) entered into a study and followed for several years.
Each child was classified as a sweetened-cereal lover or a non-sweetened
cereal lover. At the end of the study, the amount of tooth damage was
measured. Here is the summary data:
Group
Sugar Bombed
No sugar
n
10
15
mean
6.41
5.20
std. dev
5.0
15.0
An approximate 95% confidence interval for the difference in the mean
tooth damage is:
q
5
(a) (6.41 − 5.20) ± 2.26 10
+ 15
15
q
225
(b) (6.41 − 5.20) ± 2.26 25
10 + 15
q
225
(c) (6.41 − 5.20) ± 1.96 25
10 + 15
c
2006
Carl James Schwarz
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q
146
10
+
146
15
q
146
10
+
146
15
(d) (6.41 − 5.20) ± 2.07
(e) (6.41 − 5.20) ± 1.96
Solution: b
Past performance 1990 Dec - 55%
7. An experiment was conducted to compare the efficacies of two drugs in
the prevention of tapeworms in the stomachs of a new breed of sheep.
Samples of size 5 and 8 from each breed were given the drug and the two
sample means were 28.6 and 40.0 worms/sheep. From previous studies, it
is known that the variances in the two groups are 198 and 232, respectively,
and that the number of worms in the stomachs has an approximate normal
distribution. A 95% confidence interval for the the difference in the mean
number of worms per sheep is:
(a) −11.4 ± 18.6
(b) 11.4 ± 18.2
(c) −11.4 ± 17.9
(d) 11.4 ± 16.2
(e) −11.4 ± 16.6
Solution: d
Past performance 1989 Dec - 43% (27% -a)
8. A researcher wants to see if birds that build larger nests lay larger eggs.
She selects two random samples of nests: one of small nests and the other
of large nests. She weighs one egg from each nest. The data are summarized below.
sample size
sample mean (g)
sample variance
small nests
60
37.2
24.7
large nests
159
35.6
39.0
A 95% confidence interval for the difference between the average mass of
eggs in small and large nests.
(a) 1.6 ± 1.33 = (0.27, 2.93)
(b) 1.6 ± 1.48 = (0.12, 3.08)
(c) 1.6 ± 1.59 = (0.01, 3.19)
(d) 1.6 ± 1.76 = (−0.16, 3.36)
c
2006
Carl James Schwarz
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(e) 1.6 ± 7.31 = (−5.71, 8.91)
Solution: c
Past performance 1992 Dec - 82%
9. Refer to the previous question. We wish to be 95% confident of being
within 1.0 g of the true value. What is the approximate sample size is
needed for each group?
(a) 240
(b) 60
(c) 8000
(d) 2000
(e) 125
Solution: a
Past performance 1992 Dec - 79%
The following 2 questions refer to the following situation
A researcher wants to see if birds that build larger nests lay larger eggs.
She selects two random samples of nests: one of small nests and the other
of large nests. She measures one egg from each nest. The data are summarized below.
c
2006
Carl James Schwarz
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10. Refer to the 95% confidence interval circled on the output. This means:
(a) We are 95% confident that the sample mean egg size in large nests is
between 37 and 40 mm if the survey was repeated.
(b) If the survey was repeated, we are 95% confident that eggs sizes in
large nests are between 37 and 40 mm.
(c) We are 95% confident that nests will be have large eggs between 37
and 40 mm if the survey was repeated.
(d) We are 95% confident that the true mean eggs size for large nests is
between 37 and 40 mm.
(e) We are 95% confident that repeated surveys will have population
means between 37 and 40 mm.
Solution: d
Past performance 2006 Dec - 61% (19%-a; 12%-b)
11. Which of the following is NOT CORRECT?
c
2006
Carl James Schwarz
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(a) Because the 95% confidence interval for the difference in means includes zero, there is no evidence of a difference in the mean egg size.
(b) Because the one-sided p-value is .18, there is no evidence of a difference in mean egg sizes.
(c) Because the confidence intervals for the two groups have a great deal
of overlap, there is no evidence of a difference in the mean egg size.
(d) Because the individual values of the eggs sizes for the two groups
have a great deal of overlap, there is no evidence of a difference in
the means.
(e) Because the 95% confidence intervals for the mean eggs sizes are
approximately equal in width, the two estimates are about equally
precise.
Solution: d
Past performance 2006 Dec - 58% (14%-a; 15%-b; 19%-d)
c
2006
Carl James Schwarz
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