Problems: Confidence Intervals Part
2
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11. Suppose we are testing a make of car to determine its
gas mileage. We know that miles per gallon is normally
distributed with a standard deviation of one mile per
gallon. (Never mind how we know thiswe just do.)
 a) Taking a random sample of 16 cars, we find the
sample mean is 24 miles per gallon. Find the interval in
which we are 95% certain that the true mean lies.
 b) Suppose we do not know the standard deviation but
estimate it from the sample, finding that the standard
deviation of the 16 cars is one mile per gallon. Find the
interval in which we are 95% certain the true mean
lies.
12. Suppose the weights of oranges from the trees in a
grove of orange trees are normally distributed.
 a) If a random sample of sixteen is taken and the
sample mean is 3.2 ounces with a sample standard
deviation of .8 ounces, what is the interval within which
we are 90% confident that the true mean lies?
 b) Suppose that instead of using the ttables as you
should have in the previous question, you used the normal
tables. What answer would you get? Why are these answers
different?
13. Suppose an investigator wants to know the average age
of automobiles in Indiana.
 a) Would the distribution of car ages be normal,
uniform, binomial, or some other distribution? Is it a
discrete or continuous distribution?
 b) Suppose that the investigator takes a random
sample of 100 cars. What distribution will the sample
mean have? Or is the sample mean a fixed number?
Explain.
 c) What is the expected value of the sample mean? Why
is this important?
 d) Suppose the investigator finds a sample mean of
4.4 years and a sample standard deviation of 2.1 years.
(With a sample this large the sample standard deviation
is a fairly reliable estimate of the population standard
deviation and may be used in its place.) What is a 95%
confidence interval for the true average age of Indiana
cars? (Data are all hypothetical.)
14. A random sample of 64 bricks weighs 50.28 ounces with
a standard deviation of .8 ounces.
 a) What is the 95% confidence interval for the true
average weight of the bricks produced by this
process?
 b) What does this interval mean?
15. An agronomist is interested in the yield of a newly
developed tomato variety. A sample of 25 plants gives an
average yield of 34 pound during a 120day growing season,
with a sample standard deviation of three pound. What is the
95% confidence interval for the true average yield of this
variety?
16. A politician facing an upcoming election wants to
know what proportion of a population favors him over his
opponent. He hires a pollster to give him a result that is
accurate within .02 (or 2%) with 95% confidence. Assuming
that this is a close race, how large a sample must the
pollster take? (Hint: in finding the standard error of the
proportion, use p=.5)
17. Suppose a lumber company is interested in purchasing
10,000 acres of forest. If the trees have an average
diameter of 2.5 feet and a standard deviation of .5
feet,
 a) what percentage of frees will have a diameter
between two and three feet if tree diameters are normally
distributed?
 b) what percentage will have a diameter between three
and four feet?
 c) If two trees are picked at random, what is the
probability that they will both have diameters greater
than three feet?
 d) Suppose this company does not know the average
diameter of the trees, but does know that the standard
deviation is one foot. They intend to estimate the true
mean with a sample mean. How large a random sample do
they need if they want to be 95% confident that they will
estimate the true mean within three inches?
 e) How large a random sample do they need if they
want to be 95% confident that they will estimate the true
mean within one inch?
 f) The company is also interested in how many trees
are in the forest. To estimate this they select five
oneacre plots at random and count the trees on each
plot. Why is it important that the plots be selected
randomly?
 g) Suppose they get this result:
 Plot A: 40 trees
 Plot B: 46 trees
 Plot C: 36 trees
 Plot D: 39 trees
 Plot E: 39 trees
 What is the arithmetic mean of this sample?
 h) What is the median?
 i) What is the variance?
 j) What is the standard deviation?
 k) Suppose from past experience the company is
certain that the standard deviation of trees per acre in
a forest is four trees. How confident are they that there
are at least 36 trees per acre in this forest based on
the above sample?
18. The average SAT verbal score of a random sample of 25
sophomores from a large college was 394 with a standard
deviation of 75. What is:
 a) a 90% confidence interval for the entire
population of college sophomores at this college?
 b) a 95% confidence interval for the entire
population of college sophomores at this college?
 c) Approximately how much confidence would we have in
the interval 394 ± 20?
 d) How big would our sample size have to be to have a
95% confidence interval that is ± 20?
19. The following IQ scores have been obtained from a
random sample of persons from a large population: {70, 110,
110, 110, 110, 80, 120, 110, 90}. Construct a 95percent
confidence interval estimate of the population mean IQ.
20. Suppose that the average weight of widgets is 130
grams. Taking repeated samples of 25 observations, we find
that 95% of the time the sample mean will fall in the
interval 120 to 140.
 a) Approximately what is the standard deviation of
the original population?
 b) How large would our sample size have to be if we
wanted 95% of our observations to fall in the interval
125 to 135?
21. A random sample of 400 fifth grades finds that they
eat an average of 1850 calories per day. The standard
deviation of the sample is 250 calories. What is a 95%
confidence interval for the average amount of calories that
all fifth graders eat?
22. A sample of 16 light bulbs is tested to see how
bright they burn. The average of the sample is 2000 lumens
with a sample standard deviation of 48 lumens.
 a) What is the 99% confidence interval for the true
mean?
 b) What is the 90% confidence interval for the true
mean?
23. A sample of observations has been take and using
SPSS, the following results are obtained.
One Sample Statistics:



N

Mean

Std. Deviation

Std Error Mean

One

141

510.5332

252.5299

21.3511

Two

141

21.7028

6.3092

.5313



Test Value =0

t

df

sig (2tailed)

Mean Difference

95% Confidence Interval of the
Difference

Lower

Upper

One

23.911

140

.000

510.5332

468.3210

552.7454

Two

40.846

140

.000

21.7028

20.6523

22.7532

 a) Compute a 90% confidence interval for the first
variable.
 b) Compute a 99% confidence interval for the second
variable.
 c) Suppose we wanted to test the hypothesis that the
true mean of the first variable was 550. Could we reject
this if we set alpha = .05? Compute the appropriate
tvalue and explain what it tells you. (This part of the
question requires knowledge of hypothesis testing, the
topic of the next section.)
24. A test of 25 cigarette lighters show that they
average 2947 lights with a standard deviation of 250
lights.
 a) Using the ttables, construct a 95% confidence
interval for the population mean.
 b) Using the normal tables, construct a 95%
confidence interval for the population mean. How do you
explain the difference in your result from part a?
 c) Using the Normal table, construct a 98% confidence
interval for the population mean.
 d) Approximately how much confidence would you have
in the interval 2947 ± 50?
 e) How large must the sample be if want a precision
of ± 50 with 95% confidence?
26. A random sample of 100 ears of corn from a field
shows that they weigh an average of .82 pounds with a sample
standard deviation of .12 pounds.
 a) What is a 94 confidence interval for the true mean
weight?
 b) What does this answer mean?
 c) Suppose the person who took the sample and
computed the results did not know much about statistics
and he forgot to include the sample mean. Instead he only
reported that the median of the sample was .81. If the
original distribution of the weight of corn is symmetric,
the population mean and the population median are the
same, and the sample median can be used as an estimate of
the population mean. However, as we take sample after
sample, the median of the samples will bounce around more
than the meanthe sample median will have a distribution
that has a bigger standard deviation than the
distribution of the sample mean. Some brilliant
mathematician showed that the distribution of the sample
median will have a standard deviation that is larger than
the standard error of the mean by a factor of the square
root of one half of pi. What would a 94% confidence
interval of the true mean be if the sample median was
used as the estimator of the population mean? (Note: this
question is highlighting that one of the nice properties
of the mean compared to the median is that it is more
efficientthat is, it bounces around less from sample to
sample than the sample median does.)
27. Congressman Friendly mails a questionnaire to his
constituents to learn their views on an energy issue. Can he
draw valid conclusions about how all of his constituents
view this issue from the results of his survey? Explain. If
he cannot draw valid conclusions, why would he do the
survey?
28. A random sample of 1600 people is taken to obtain
their view on the economy. 67% say they expect things to get
worse. What is a 95% confidence interval for the entire
population?
29. A garden seed distributor wants to test the
germination of a shipment of potato seeds. (Excuse the
attempted humor.) Previous experience has shown that he can
expect between 75% and 98% of them to germinate. How large a
sample must he take if he wants to estimate the percentage
of this shipment that will germinate within ± .05 with
05% confidence?
30. A garden seed distributor wants to test the
germination rate of a shipment of onion seed. He has no idea
of what the rate may be. How large a sample must he take if
he wants to estimate the percentage of this shipment that
will germinate within ± .05 with 95% confidence?
31. Suppose that a retail store is interested in knowing
the average age of their inventory. They take a random
sample of 400 items and find that the average age is 98 days
with a standard deviation of 38 days.
 a) What is a 90% confidence interval for the
population mean in this case?
 b) Suppose that they found that the sample median was
96 days. Find a 90% confidence interval for the
population mean using the sample median as a point
estimate instead of the sample mean. (For this we need to
assume that the original population was distributed as a
normal distribution, in which case the mean and median
are identical. However, the standard error of the sample
median will be larger than the standard error of the
standard error of the sample mean by a factor of 1.2533,
which is the square root of onehalf of pi.)
 c) If two estimators are unbiased (they tend not to
overpredict or underpredict), the one whose sampling
distribution has the smaller variance is more efficient.
The sample median is an unbiased estimator of the sample
mean if the original population is distributed normally,
but as mentioned in part b, it has a larger standard
deviation. What must its distribution look like compared
to the distribution of the sample mean? (Assume that the
curve below shows the distribution of the sample mean
form sample of sample. draw on an appropriate curve
showing what the distribution of the sample median would
look like.)
 d) Both the sample mean and the sample median are
consistent estimators of the sample mean if the original
distribution is normal. Loosely speaking, an estimator is
consistent if for large samples the probability is close
to one that the estimator will be near the population
parameter being estimated. Explain why they are
consistent. (Hint: what happens to the standard error of
the sample mean as n gets larger and larger?)
