Download Exam #2 - TAMU Stat

STAT303: Secs 509-511
Spring 2003
Exam #2
Form A
Instructor: Julie Hagen Carroll
1. Don’t even open this until you are told to do so.
2. Be sure to mark your section number (509, 510 or 511) on the scantron!
3. There are 20 multiple-choice questions on this exam, each worth 5 points. There is partial credit. Please
mark your answers clearly on the scantron. Multiple marks will be counted wrong.
4. You will have 60 minutes to finish this exam.
5. If you are caught cheating or helping someone to cheat on this exam, you both will receive a grade of
zero on the exam. You must work alone.
6. This exam is worth 100 points, and will constitute 25% of your final grade.
7. Good luck!
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STAT303: 509-511
Exam #2, Form A
D. How likely are you to get a sample proportion of 30% or more if you take a sample of
40 from a population with mean 25%?
E. Math doesn’t equate to words.
1. A statistic is an unbiased estimator if
A. the original population from which the sample is drawn is normal.
B. the sampling distribution of the statistic is
normal.
C. the mean of the sampling distribution of the
statistic is the parameter of interest.
D. it has the smallest variance of all the estimators.
E. the mean of every possible sample of size n
from the population is µ.
5. “Bee pollen is effective for combating fatigue, depression, cancer and colon disorders.” So says a
Web site that offers the pollen for sale. We wonder if the bee pollen really does prevent colon
disorders. Which of the following would most
accurately determine the effect of bee pollen on
preventing colon disorders?
A. Take a SRS of people who have colon disorders and find the proportion that take bee
pollen.
B. Take a SRS of people who take bee pollen
and find the proportion that have colon disorders.
C. Take a SRS of people who do not have colon
disorders. Randomly assign half of the sample to take bee pollen pills and the other
half to take placebos. Have both groups
take the pills regularly for five years and
compare the proportion of each group that
have colon disorders.
D. Take a SRS from both the population of
people who take bee pollen and a SRS from
the population of people who do not take
bee pollen. Compare the proportion of people in both samples who have colon disorders.
E. Any of the above would work.
2. The phrase (1 − α) ∗ 100% confidence means
A. the probability of the confidence interval
NOT capturing the value of the population
parameter is α.
B. for a large enough sample size n, we can be
(1 − α) ∗ 100% confident of capturing the
true value of µ.
C. for a large enough sample size n, our
method of constructing confidence intervals
will be correct (1 − α) ∗ 100% of the time.
D. the estimator X is unbiased for (1 − α) ∗
100% of the possible values of X based on
all possible sample of size n, provided n is
large enough.
E. if we could take all possible samples of
size n and calculate the (1 − α) ∗ 100% confidence interval for each sample, then exactly (1−α)∗100% of these intervals would
contain the value of the population parameter.
6. If we each generated 20 random samples from
a population of N (4, 102 ). From these samples,
we created 20 95% confidence intervals. If we
looked at all of our confidence intervals collectively (there were about 400), then
3. Let X 4 ∼ N (20, 52 ). What is P (18 < X 4 < 22)?
A.
B.
C.
D.
E.
Spring 2003
0.5762
0.5
0.3108
0
practically 1
A. it is plausible that only 370 of them actually
contained 4.
B. although it’s not very likely, it is plausible
that all of them actually contained 4.
C. all of them would contain 4 since we all
did the same thing and we KNOW the true
mean is 4.
D. 380 of them would contain 4.
E. Exactly two of the statements above are
plausible.
4. Which statement agrees with P (p40 > 0.30) for
p40 ∼ N (0.25, 0.0682 )?
A. How likely are you to get a sample proportion of 30% if you take a sample of 40 from
a population with mean 25%?
B. How likely are you to get 30% or more if
you sample a population of 40?
C. How likely are you to get a probability
greater than 0.30 if you sample a population with mean 0.25 and standard deviation
0.068?
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STAT303: 509-511
Exam #2, Form A
7. Suppose the true mean and standard deviation
of Duracell Alkaline AA battery lifetime are 5.1
hr and 1.8 hr, respectively. Those of Eveready
batteries are 4.9 hr and 2.1 hr, respectively.
If D100 is the sample mean of 100 Duracells
and E 100 is the sample mean of 100 Evereadys,
what is the approximate sampling distribution of
D100 − E 100 ?
A.
B.
C.
D.
E.
11. Referring to the confidence interval in the previous question, a 99% confidence interval for the
mean weight of high school girls in pounds is
(102.3, 106.5), which of the following is true?
A. Approximately 99% of high school girls
weigh between 102.3 and 106.5 pounds.
B. There is a 99% probability that the true
mean weight of high school girls is 104.4
and 99% of the time the margin of error
will be 2.1.
C. If we repeatedly sampled the weight of high
school girls, 99% of the sample means would
be between 102.3 and 106.5.
D. If we repeatedly sampled the weight of high
school girls, 99% of the time the true mean
would be between 102.3 and 106.5.
E. If we repeatedly sampled the weight of high
school girls, 99% of the time the true mean
fall in the calculated interval.
N (0.2, 0.32 )
N (0.2, 0.2772 )
N (0.2, 0.032 )
N (0.2, 0.392 )
N (2, 3.92 )
8. Which of the following is true?
A. The proportion of times a particular event
A occurs in many, many repetitions is approximately the probability of the event A.
B. As you keep repeating the same experiment,
the proportion of times a particular event A
occurs will approach the true probability of
the event A.
C. The law of large numbers says that the distribution of the sample proportions will approach the normal distribution as you keep
repeating the same experiment.
D. All of the above are true.
E. Exactly two of the above are true.
12. Z ∼ N (0, 12 ). What is ±zα/2 for a 55% confidence interval? Find ±zα/2 such that P (−zα/2 <
Z < zα/2 ) = 0.55?
A. ±0.55
B. ±0.75
C. ±0.13
D. −0.29 and 0.71
E. ±0.59
9. Suppose X ∼ N (10, 142 ). How large of a sample would you need to take to have the standard
deviation of the sample mean to be half as big?
A.
B.
C.
D.
E.
13. Suppose you know the true population proportion, π = 0.85. How large of a sample will you
need to take if you want to use the normal approximation to estimate probabilities?
at least 30
28
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4
56
A. We need to know how small we want the
margin of error before we can answer this
question.
B. at least 30
C. 12 is enough
D. at least 67
E. at least 85
10. Suppose a 99% confidence interval for the true
mean weight of high school girls in pounds is
(102.3, 106.5). If we had measured the weights
of each of the girls in kilograms (2.2 pounds =
1 kilogram) then the confidence interval for the
mean weight of high school girls in kilograms
would have been
A.
B.
C.
D.
E.
Spring 2003
14. Let X be a continuous uniform random variable
like a problem in your homework, so the range
of X is 0 to 10 and the mean is 5. What is
P (X = 5)?
(104.5, 106.7).
(46.5, 48.4).
(225.06, 234.3).
(100.1, 104.3).
indeterminable. We would have to regather
the data.
A.
B.
C.
D.
E.
3
0 because 5 is the mean
0.5 because 5 is the mean
0.5 because 5 is the median
0.5 because X is continuous
0 because X is continuous
STAT303: 509-511
Exam #2, Form A
15. Suppose a 90% confidence interval for the true
proporion of A&M students who watch Walker:
Texas Ranger is (0.22, 0.53), and a 90% confidence interval for the true proportion of t.u.
students who watch is (0.29, 0.42). From this we
can conclude
19. The standard normal distribution
A. always has the mean equal to the median
equal to zero.
B. always has the standard deviation equal to
the IQR and equal to 1.
C. always has the smallest standard deviation
of all normal distributions.
A. more Aggies watch than t.u. students.
B. less Aggies watch than t.u. students.
C. it’s plausible that the same proportion from
either school watch.
D. we have more confidence that the A&M interval contains the true proportion since it
is wider than the other.
E. t.u. students are not as patriotic as Aggies.
D. All of the above are true.
E. Only two of A, B and C are true.
20. The level of confidence is compromised when the
population from which a sample is taken is not
normally distributed. This implies
A. although we state a level of confidence equal
to (1−α)100∗%, the true level of confidence
may not be (1 − α) ∗ 100%, but something
quite different.
B. although we say we are calculating a 95%
confidence interval, we’re really only calculating a 68% confidence interval.
C. if the sample size is NOT large enough, the
distribution of the confidence intervals will
NOT be symmetric.
D. unless the sample size is large enough, we
will get confidence intervals that are too
wide.
E. the confidence intervals will be biased.
16. What is the 96th percentile of X ∼ N (2, 42 )?
A.
B.
C.
D.
E.
Spring 2003
0.8289
1.75
5.32
9
5.84
17. The textbook talks about many different ways
to sample a population. The ‘good’ ones always
use some form of random sampling. Why?
A. A random sample causes the mean of the
sample means, µx̄ , to equal the mean of the
population, µx .
B. A random sample causes the standard de- 1C,2E,3C,4D,5C,6E,7B,8E,9D,10B,11E
viation of the sample means, σx̄ , to equal 12B,13D,14E,15C,16D,17E,18B,19A,20A
the standard deviation of the population divided by the square root of the sample size,
σx
√
.
n
C. A random sample causes the distribution of
the sample means to be normal.
D. All of the above are true.
E. Exactly two of the above are true.
18. Bonnie is trying to get a reasonably narrow confidence interval for the true proportion of Aggies who support Bonfire. She determines her
interval is not NARROW enough. Which of the
following could help her make her confidence interval narrower?
A.
B.
C.
D.
decreasing her sample size
decreasing her confidence level
decreasing her sample mean
All of the above would decrease the width
of her interval.
E. None of the above would decrease the width
of her interval.
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