Download Chapter 5 and Chapter 6 Review READ: Here are some problems I

Chapter 5 and Chapter 6 Review
READ: Here are some problems I selected for the review. This does not mean that only the concepts or
type of problems on this review will only be on the test. Make sure you go over your homework and study
the PowerPoints and notes.
Key Terms
Interval Estimator, Normal (z) statistic, Point Estimator, Sampling error, T-statistic, Degrees of Freedom
(df), Target Parameter, Confidence interval, Confidence Level, Rejection Region, Alternative hypothesis,
Null Hypothesis, Type I error, Type II error, Two-tailed Hypothesis, One-Tailed Hypothesis, Observed
significance, Conclusion
Understanding the Principles
1. Explain the difference between a one-tailed and a two-tailed test.
2. What is the difference between Type I and Type II errors in hypothesis testing? How does 𝛼 and
𝛽 relate to Type I and Type II error?
3. List the four possible results of the combinations of decisions and true states of nature for a test of
hypothesis.
4. Interpret the phrases “95% confident” in the following statement: “We are 95% confident that the
proportion of all PCs with a computer virus falls between .12 and .18.”
Practice Problems
1.
Let 𝑡0 represent a particular value of t from Table IV. Find the table values such that the
following statements are true:
a. 𝑃(𝑡 ≥ 𝑡0 ) = .005, where df = 14
b. 𝑃(𝑡 ≤ −𝑡0 𝑜𝑟 𝑡 ≥ 𝑡0 ) = .10, where df = 6
2. A random sample of 225 measurements is selected from a population, and the sample mean and
standard deviation are 𝑥̅ = 32.5 and s = 30.0, respectively.
a. Use a 99% confidence interval to estimate the mean of the population, 𝜇.
b. How large a sample would be needed to estimate 𝜇 to within .5 with 99% confidence?
c. What is meant by the phrases “99% confidence” as it is mused in this exercise?
3. According to a Food and Drug Administration (FDA) study, a cup of coffee contains an average
of 115 milligrams (mg) of caffeine, with the amount per cup ranging from 60 to 180 mg. Suppose
you want to repeat the FDA experiment in order to obtain an estimate of the mean caffeine
content in a cup of coffee correct to within 5 mg with 95% confidence. How many cups of coffee
would have to be included in your sample?
4. Find 𝑧𝛼/2 for each of the following.
a. 𝛼 = .01
b. 𝛼 = .30
5. The mean and standard deviation of a random sample of n measurements are equal to 33.9 and
3.3, respectively.
a. Find a 95% confident interval for µ if n = 100.
b. Find a 95% confident interval for µ if n = 400.
c. Find the widths of the confidence intervals you calculated in parts a and b. What is the
effect on the width of a confidence interval of quadrupling the sample size while holding
the confidence coefficient fixed?
6. Strategic placement of lobster traps is one of the keys for a successful lobster fisherman. An
observational study of teams fishing for the red spiny lobster in Baja California Sur, Mexico, was
conducted and the results published in Bulletin of Marine Science (April, 2010). One of the
variables of interest was the average distance separating traps—called trap spacing – deployed by
the same team of fisherman. Trap spacing measurement (in meters) for a sample of seven teams
of red spiny lobster fisherman are shown below in the table. Let 𝜇 represent the average of the
trap spacing measurements for the population of red spiny lobster fishermen fishing in Baja
California Sur, Mexico. Suppose you want to determine if the true value of 𝜇 differs from 95
meters.
93
99
105
94
82
70
86
a.
b.
c.
d.
Specify the null and alternative hypothesis for this test.
Computer the value of the test statistic.
Find the approximate p-value of the test.
Select a value of 𝛼, the probability of a Type I error. Interpret this value in the words of the
problem.
e. Give the appropriate conclusion, based on the results of parts c and d.
7. A key statistic used by football coaches to evaluate players is a player’s 40-yard sprint time. Can
a drill be developed for improving a player’s speed in the sprint? Researchers at Northern
Kentucky University designed and tested a speed-training program for junior varsity and varsity
high school football players (The Sport Journal, Winter 2004). The training program included 50yard sprints run at a varying speeds, high knee running sprints, butt kick sprints, “crazy legs”
straddle runs, quick fee drills, jumping, power skipping and all-out sprinting. Each in a sample of
38 high school athletes was timed in a 40-yard sprint prior to the start of the training program and
timed again after completing the program. The decreases in times (measured in seconds) are
listed in the table below. [Note: A negative decrease implies that the athlete’s time after
completion of the program was higher than his time prior to training.] The goal of the research is
to demonstrate that the training program is effective in improving 40-yard sprint times.
-.01
.32
.15
.01
.1
.34
.3
.9
.1
.3
.02
.34
.24
.09
.12
.38
.25
.05
.14
.44
.05
0
.1
.08
.28
.04
.08
0
.25
.17
.5
0
.2
0
.36
.14
.21
.1
a. Find a 95% confidence interval for the true mean decrease in sprint times for the
population of all football players who participate in the speed-training program.
b. Based on the confidence interval, is the training program really effective in improving the
mean 40-yard sprint time of high school football players? Explain.
8. Let 𝑡0 be a particular value of t. Use Table IV to find 𝑡0 values such that the following statements
are true:
a. 𝑃(−𝑡0 < 𝑡 < 𝑡0 ) = .95, where df = 16
b. 𝑃(𝑡 ≤ −𝑡0 𝑜𝑟 𝑡 ≥ 𝑡0 ) = .10, where df = 12
9.
Three-way catalytic converters have been installed in new vehicles in order to reduce pollutants
from motor vehicle exhaust emissions. However, these converters unintentionally increase the
level of ammonia in the air. Environmental Science & Technology (Sept. 1, 2000) published a
study on the ammonia levels near the exit ramp of a San Francisco highway tunnel. The ammonia
concentration (parts per million) data for eight randomly selected days during the afternoon drive
time are reproduced in the table below. Find a 99% confidence interval for the population mean
daily ammonia level in air in the tunnel. Interpret your result.
1.53
1.50
1.37
1.51
1.55
1.42
1.41
1.48
10. “The Training Game” is an activity used in psychology in which one person shapes an arbitrary
behavior by selectively reinforcing the movements of another person. Students at Georgia
Institute of Technology played the game at Zoo Atlanta while participating in an experimental
psychology laboratory in which they assisted in the training of animals (Teaching of Psychology,
May 1998). A sample of 15 psychology students rated “The Training Game” as a “great” method
of understanding the animal’s perspective during training of 7-1 point scale (where 1 = strongly
disagree and 7 = strongly agree). The mean response was 5.87, with a standard deviation of 1.51.
a. Construct a 95% confidence interval for the true mean response of the students.
b. Suppose you want to reduce the width of the 95% confidence interval to half the size
obtained in part a. How many students are required in the sample in order to obtain the
desired confidence interval width?
11. A random sample of 175 measurements possessed a mean of 𝑥̅ = 8.2 and standard deviation of
s = .79.
a. Form a 95% confidence interval for 𝜇.
b. Test 𝐻𝑜 : 𝜇 = 8.3 against 𝐻𝑎 : 𝜇 ≠ 8.3. Use 𝛼 = .05.
12. Medical tests have been developed to detect many serious diseases. A medical test is designed to
minimize the probability that it will produce a “false positive” or a “false negative.” A false
positive is a positive test result for an individual who does not have the disease, where as a false
negative is a negative test result for an individual who does have the disease.
a. If we treat a medical test for a disease as a statistical test of hypothesis, what are the null
and alternative hypotheses for the medical test?
b. What are the Type I and Type II errors for the test? Relate each to false positives and
false negatives.
c. Which of these errors has graver consequences? Considering this error, is it more
important to minimize 𝛼 or 𝛽? Explain.
13. Refer to the Journal of Forest Engineering (July 1999) study on minimizing tractor skidding
distances along a new road in a European forest. The skidding distances (in meters) were
measured at 20 randomly selected road sites. The data is represented in the table below. The
logger working on the road claims that the mean skidding distance is at least 425 meters. Is there
sufficient evidence to refute this claim? Use 𝛼 = .10. Stat the appropriate conclusion in the words
of the problem.
488
385
350
295
457
184
199
261
285
273
409
400
435
311
574
312
439
141
546
425
14. Health care workers who use latex gloves with glove powder on a daily basis are particularly
susceptible to developing a latex allergy. Symptoms of a latex allergy include conjunctivitis, hand
eczema, nasal congestion, a skin rash, and shortness of breath. Each in a sample of 46 hospital
employees who were diagnosed with latex allergy based on a skin-prick test reported on their
exposure to latex gloves (Current Allergy & Clinical Immunology, March 2004). The number of
latex gloves used per week by the sampled workers is summarized as follows: 𝑥̅ = 19.3 and s =
11.9. Let 𝜇 represent the mean number of latex gloves used per week by all hospital employees.
Consider testing 𝐻0 : 𝜇 = 20 against 𝐻𝑎 : 𝜇 < 20.
a. Give the rejection region for the test at a significance level of 𝛼 = .01.
b. Calculate the value of the test statistic.
c. Use the results from parts a and b to draw the appropriate conclusion.
d. Compute the p-value of the test.
e. Compare the p-value with 𝛼 = .01and make the appropriate conclusion.