|
|
Exercises
Author(s)
David M. Lane
Prerequisites
All material presented in the Estimation Chapter
Selected answers
You may want to use the Analysis Lab and various calculators for some of these exercises.
Calculators:
Inverse t Distribution: Finds t for a confidence interval.
t Distribution: Computes areas of the t distribution.
Fisher's r to z': Computes transformations in both directions.
Inverse Normal Distribution: Use for confidence intervals.
- When would the mean grade in a class on a final exam be considered
a statistic? When would it be considered a parameter?
(relevant section)
- Define bias in terms of expected value.
(relevant section)
- Is it possible for a statistic to be unbiased yet very imprecise?
How about being very accurate but biased? (relevant
section)
- Why is a 99% confidence interval wider than a 95% confidence
interval? (relevant section & relevant
section)
- When you construct a 95% confidence interval, what are you
95% confident about?
(relevant section)
- What is the difference in the computation of a confidence
interval between cases in which you know the population standard
deviation and cases in which you have to estimate it?
(relevant section & relevant
section)
- Assume a researcher found that the correlation between a
test he or she developed and job performance was 0.55 in a
study of 28 employees. If correlations under .35 are considered
unacceptable, would you have any reservations about using this
test to screen job applicants? (relevant
section)
- What is the effect of sample size on the width of a confidence
interval? (relevant section & relevant
section)
- How does the t distribution compare with the normal distribution?
How does this difference affect the size of confidence intervals
constructed using z relative to those constructed using t?
Does sample size make a difference? (relevant
section)
- The effectiveness of a blood-pressure drug is being investigated.
How might an experimenter demonstrate that, on average, the
reduction in systolic blood pressure is 20 or more? (relevant
section & relevant
section)
- A population is known to be normally distributed with a standard
deviation of 2.8. (a) Compute the 95% confidence interval on
the mean based on the following sample of nine: 8, 9, 10, 13,
14, 16, 17, 20, 21. (b) Now compute the 99% confidence interval
using the same data. (relevant section)
- A person claims to be able to predict the outcome of flipping
a coin. This person is correct 16/25 times. Compute the 95%
confidence interval on the proportion of times this person
can predict coin flips correctly. What conclusion can you draw
about this test of his ability to predict the future? (relevant
section)
- What does it mean that the variance (computed by dividing
by N) is a biased statistic? (relevant
section)
- A confidence interval for the population mean computed from
an N of 16 ranges from 12 to 28. A new sample of 36 observations
is going to be taken. You can't know in advance exactly what
the confidence interval will be because it depends on the random
sample. Even so, you should have some idea of what it will
be. Give your best estimation. (relevant
section)
- You take a sample of 22 from a population of test scores,
and the mean of your sample is 60. (a) You know the standard
deviation of the population is 10. What is the 99% confidence
interval on the population mean. (b) Now assume that you do
not know the population standard deviation, but the standard
deviation in your sample is 10. What is the 99% confidence
interval on the mean now? (relevant section)
- You read about a survey in a newspaper and find that 70%
of the 250 people sampled prefer Candidate A. You are surprised
by this survey because you thought that more like 50% of the
population preferred this candidate. Based on this sample,
is 50% a possible population proportion? Compute the 95% confidence
interval to be sure. (relevant
section)
- Heights for teenage boys and girls were calculated.
The mean height for the sample of 12 boys was 174 cm and the
variance was 62. For the sample of 12 girls, the mean was 166
cm and the variance was 65. (a) What is the 95% confidence
interval on the difference between population means? (b) What
is the 99% confidence interval on the difference between population
means? (c) Do you think the mean difference in the population
could be about 5? Why or why not? (relevant
section)
- You were interested in how long the average psychology major
at your college studies per night, so you asked 10 psychology
majors to tell you the amount they study. They told you the
following times: 2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4. (a) Find
the 95% confidence interval on the population mean. (b) Find
the 90% confidence interval on the population mean. (relevant
section)
- True/false: As the sample size gets larger, the probability
that the confidence interval will contain the population mean
gets higher. (relevant section & relevant
section)
- True/false: You have a sample of 9 men and a sample of 8
women. The degrees of freedom for the t value in your confidence
interval on the difference between means is 16. (relevant
section & relevant
section)
- True/false: Greek letters are used for statistics as opposed
to parameters. (relevant section)
- True/false: In order to construct a confidence interval on
the difference between means, you need to assume that the populations
have the same variance and are both normally distributed. (relevant
section)
- True/false: The red distribution represents the t distribution
and the blue distribution represents the normal distribution.
(relevant section)
Questions from Case Studies:
The following questions are from the Angry
Moods (AM) case study.
- (AM#6c) Is there a difference in how
much males and females use aggressive behavior to improve an
angry mood? For the "Anger-Out" scores, compute a 99% confidence
interval on the difference between gender means. (relevant
section)
- (AM#10) Calculate the 95% confidence interval
for the difference between the mean Anger-In score for the athletes
and non-athletes. What can you conclude? (relevant
section)
- Find the 95% confidence interval on the population correlation
between the Anger-Out and Control-Out scores. (relevant
section)
The following questions are from the Flatulence (F)
case study.
- (F#8) Compare men and women on the variable "perday." Compute
the 95% confidence interval on the difference between means.
(relevant section)
- (F#10) What is the 95% confidence interval of the
mean time people wait before farting in front of a romantic
partner. (relevant section)
The following
questions use data from the Animal
Research (AR) case study.
- (AR#3) What percentage of the
women studied in this sample strongly agreed (gave a rating
of 7) that using animals for research is wrong?
- Use the proportion
you computed in #29. Compute the 95% confidence interval on
the population proportion of women who strongly agree that
animal research is wrong. (relevant
section)
- Compute a 95% confidence interval on the difference between
the gender means with respect to their beliefs that animal
research is wrong. (relevant section)
The following question is from the ADHD
Treatment (AT) case study.
- (AT#8) What is the correlation
between the participants' correct number of responses after
taking the placebo and their correct number of responses after
taking 0.60 mg/kg of MPH? Compute the 95% confidence interval
on the population correlation. (relevant
section)
The following question is from the Weapons
and Aggression (WA) case study.
- (WA#4) Recall that the hypothesis
is that a person can name an aggressive word more quickly if
it is preceded by a weapon word prime than if it is preceded
by a neutral word prime. The first step in testing this hypothesis
is to compute the difference between (a) the naming time of
aggressive words when preceded by a neutral word prime and
(b) the naming time of aggressive words when preceded by a
weapon word prime separately for each of the 32 participants.
That is, compute an - aw for each participant.
- Would the hypothesis of this study be supported if the
difference were positive or if it were negative?
- What is the mean of this difference score? (relevant section)
- What is the standard deviation of this difference score?
(relevant section)
- What is the 95% confidence interval of the mean difference score? (relevant section)
- What does the confidence interval computed in (d) say
about the hypothesis.
The following question is from the Diet
and Health (WA)
case study.
- Compute a 95% confidence interval on the proportion of people
who are healthy on the AHA diet.
|
Cancers
|
Deaths
|
Nonfatal illness
|
Healthy
|
Total
|
|
AHA
|
15
|
24
|
25
|
239
|
303
|
|
Mediterranean
|
7
|
14
|
8
|
273
|
302
|
Total |
22
|
38
|
33
|
512
|
605
|
The
following questions are from (reproduced with permission)
Visit the site
- Suppose that you take a random sample of 10,000 Americans
and find that 1,111 are left-handed. You perform a test of
significance to assess whether the sample data provide evidence
that more than 10% of all Americans are left-handed, and you
calculate a test statistic of 3.70 and a p-value of .0001.
Furthermore, you calculate a 99% confidence interval for the
proportion of left-handers in America to be (.103,.119). Consider
the following statements: The sample provides strong evidence
that more than 10% of all Americans are left-handed. The sample
provides evidence that the proportion of left-handers in America
is much larger than 10%. Which of these two statements is the
more appropriate conclusion to draw? Explain your answer based
on the results of the significance test and confidence interval.
- A student wanted to study the ages of couples applying for
marriage licenses in his county. He studied a sample of 94
marriage licenses and found that in 67 cases the husband was
older than the wife. Do the sample data provide strong evidence
that the husband is usually older than the wife among couples
applying for marriage licenses in that county? Explain briefly
and justify your answer.
- Imagine that there are 100 different researchers each studying
the sleeping habits of college freshmen. Each researcher takes
a random sample of size 50 from the same population of freshmen.
Each researcher is trying to estimate the mean hours of sleep
that freshmen get at night, and each one constructs a 95% confidence
interval for the mean. Approximately how many of these 100
confidence intervals will NOT capture the true mean?
- None
- 1 or 2
- 3 to 7
- about half
- 95 to 100
- other
Answers:
11) (a) (12.39, 16.05)
12) (.43, .85)
15) (b) (53.96, 66.04)
17) (a) (1.25, 14.75)
18) (a) (1.45, 3.05)
26) (-.713, -.414)
27) (-.98, 3.09)
29) 41%
33) (b) 7.16
Please answer the questions:
|
|