Chapter 10: Hypothesis Testing Using a Single Sample
10.1: Hypotheses and Test Procedures
What is a test of hypotheses?
For example: Is the value of the sample statistic…
Hypothesis statements:
Null hypothesis:
Alternative hypothesis:
Consider a murder trial. What is the null hypothesis?
What is the alternative hypothesis?
If the jury rejects the null hypothesis:
If the jury fails to reject the null hypothesis:
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The form of Hypotheses:
Null hypothesis
Alternative hypothesis
Sharing prescription drugs with others can be dangerous. A survey of a representative sample of 592
U.S. teens age 12 to 17 reported that 118 of those surveyed admitted to having shared a prescription
drug with a friend. Is this sufficient evidence that more than 10% of teens have shared prescription
medication with friends?
What is the population characteristic of interest?
What is the hypothesized value?
State the hypotheses:
Compact florescent (cfl) light bulbs are
much more energy efficient than regular incandescent light
bulbs. Ecobulb brand 60-watt cfl light bulbs state on the package “Average life 8000 hours”. People
who purchase this brand would be unhappy if the bulbs lasted less than 8000 hours. A sample of
these bulbs will be selected and tested.
What is the population characteristic of interest?
State the hypotheses:
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Because in variation of the manufacturing process, tennis balls produced by a particular machine do
not have the same diameters. Suppose the machine was initially calibrated to achieve the
specification of m = 3 inches. However, the manager is now concerned that the diameters no longer
conform to this specification. If the mean diameter is not 3 inches, production will have to be halted.
What is the population characteristic of interest?
State the hypotheses:
For each pair of hypotheses, indicate which are not legitimate and explain why:
a) H 0 :   15 ; H a :   15
b) H 0 : x  4 ; H a : x  4
c) H 0 : p  .1; H a : p  .1
d) H 0 :   2.3 ; H a :   3.2
e) H0 : p  .5 ; Ha : p  .5
Practice Problems:
1. Criminal Trials: The jury is always told that the defendant is “innocent until proven guilty”.
a. What must a member of the jury assume about the defendant at the beginning of the trial? H0=
b. It is the prosecuting attorney’s job to present evidence to the juty. If there is enough evidence,
them the jury will convict the defendant of the crime. If the defendant is convicted, the jury is
rejecting the null hypothesis. Ha=
c. When the jury convicts someone of a crime, their verdict is GUILTY. Is this “Reject H0” or “Fail to
Reject H0”?
d. If the jury fails to convict someone of a crime, their verdict is NOT GUILTY. Is this “Reject H 0” or
“Fail to Reject H0”?
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e. How does the verdict of “not guilty” differ from “innocent”?
f. Sometimes the jury makes a correct decision and sometimes the jury makes a mistake. When H 0 is
true, but we reject it based on the sample evidence, this is an error. We call it a Type I error. Write
a sentence describing a Type I error in the U.S. criminal justice system.
g. When H0 is false, but we fail to reject it based on the sample evidence, this is also an error. We call
it a Type II error. Write a sentence describing a Type II error in the U.S. criminal justice system.
Recommended Homework: Read 10.1; Problems 2, 4, 5, 6-10
10.2: Errors in Hypothesis Testing
When you perform a hypothesis test you make a decision:
Type I Error
Type II Error
𝐻0 is true
𝐻0 is false
Reject 𝐻0
Fail to reject 𝐻0
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The U.S. Bureau of Transportation Statistics reports that for 2009 72% of all domestic passenger
flights arrived on time (meaning within 15 minutes of its scheduled arrival time). Suppose that an
airline with a poor on-time record decides to offer its employees a bonus if, in an upcoming month, the
airline’s proportion of on-time flights exceeds the overall 2009 industry rate of .72.
State the hypothesis:
State a Type I Error in Context:
State a Type II Error in Context:
In 2004, Vertex Pharmaceuticals, a biotechnology company, issued a press release announcing that it
had filed an application with the FDA to begin clinical trials on an experimental drug VX-680 that
had been found to reduce the growth rate of pancreatic and colon cancer tumors in animal studies.
Let µ= the true mean growth rate of tumors for patients taking the experimental drug
H0:
Ha:
State a Type I Error in Context:
State a Type II Error in Context:
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The relationship between α and β
Selecting a significance level α=.05 results in a test procedure that, used over and over with different
samples, reject a true H0 about 5 times in 100.
The relationship between α and β
How does one decide what α level to use?
The EPA has adopted what is known as the Lead and Copper Rule, which defines drinking water as
unsafe if the concentration of lead is 15 parts per billion (ppb) or greater or if the concentration of
copper is 1.3 ppb or greater. The manager of a community water system might use lead level
measurements from a sample of water specimens to test the following hypotheses:
State a Type I Error in Context:
State a Type II Error in Context:
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Practice Problems:
2. Medical tests have been developed to detect many serious diseases (such as cancer and HIV). A
medical test is designed to give correct results as often as possible. That is, to minimize the
occurrence of “false positives” and “false negatives”. A doctor starts by assuming that a patient is
healthy (no disease), then looks for evidence to contradict that assumption. If the patient has a
negative test result, the doctor continues to assume that the patient is healthy. If the patient has
a positive test result, the doctor concludes that patient has a disease.
a. State H0 and Ha.
b. When will the doctor reject H0?
c. When will the doctor fail to reject H0?
d. What kind of error is a “false positive”? Explain.
e. What kind of error is a “false negative”? Explain.
f.
What are the consequences of a false positive and of a false negative? Which do you think doctors
would “rather” have? What consequences might this have on the way medicine is practiced?
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3. A movie critic claims that, among children’s movies that show the use of tobacco, the mean
exposure time is less than 2 minutes.
a. Identify the population type and describe the population characteristic in words.
b. State H0 and Ha.
c. Describe a Type I error in context.
d. Describe a Type II error in context.
4. A researcher claims that over 1% of the people who take drug Lipitor experience flu-like
symptoms.
a. Identify the population type and describe the population characteristic in words.
b. State H0 and Ha.
c. Describe a Type I error in context.
d. Describe a Type II error in context.
Suggested Homework: Read 10.2; Problems 12-14, 17, 20, 22
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10.3 Large Sample Hypothesis Test for a Population Proportion
The fundamental idea behind hypothesis testing is:
Recall the General Properties for Sampling Distributions of 𝑝̂
1.
2.
3.
These three properties imply:
In June 2006, an Associated Press survey was conducted to investigate how people use the nutritional
information provided on food packages. Interviews were conducted with 1003 randomly selected adult
Americans, and each participant was asked a series of questions, including the following two:
Question 1: When purchasing packaged food, how often do you check the nutritional labeling on the
package?
Question 2: How often do you purchase food that is bad for you, even after you’ve checked the
nutrition labels?
It was reported that 582 responded “frequently” to the question about checking labels and 441
responded “very often” or “somewhat often” to the question about purchasing bad foods even after
checking the labels.
Based on these data, is it reasonable to conclude that a majority of adult Americans frequently check
nutritional labels when purchasing packaged foods?
H0:
Ha:
p=
𝑝̂ :
This observed sample proportion is greater than .5. Is it plausible a sample proportion of p = .58
occurred as a result of chance variation, or is it unusual to observe a sample proportion this large
when p = .5?
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Create a test statistic using:
A test statistic indicates how many standard deviations the sample statistic (𝑝̂ ) is from the population
characteristic (p).
z=
The P-value is the probability of obtaining a test statistic at least as inconsistent with H0 as was
observed, assuming H0 is true.
P-value≈
Conclusion:
Computing P-values
The calculation of the P-value depends on the form of the inequality in the alternative hypothesis.
Ha: p> hypothesized value
Ha: p<hypothesized value
Ha: p≠ hypothesized value
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Using P-values to make a decision:
To decide whether or not to reject H0, we compare the P-value to the significance level α:
Summary of the Large Sample z Test for p
Null hypothesis:
Test Statistic:
Alternative Hypothesis:
P-Value
Assumptions:
1.
2.
3.
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A report states that nationwide, 61% of high school graduates go on to attend a two-year or four-year
college the year after graduation. Suppose a random sample of 1500 high school graduates in 2009
from a particular state estimated the proportion of high school graduates that attend college the year
after graduation to be 58%. Can we reasonably conclude that the proportion of this state’s high
school graduates in 2009 who attended college the year after graduation is different from the national
figure? Use a = .01.
State the hypotheses:
Assumptions:
Test Statistic:
P-value:
Conclusion:
Potential Error:
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In December 2009, a county-wide water conservation campaign was conducted in a particular county.
In 2010, a random sample of 500 homes was selected and water usage was recorded for each home in
the sample. Suppose the sample results were that 220 households had reduced water consumption.
The county supervisors wanted to know if their data supported the claim that fewer than half the
households in the county reduced water consumption.
State the Hypotheses:
𝑝̂ :
Assumptions:
Test Statistic and P-value:
Conclusion:
Potential Error:
Confidence Interval:
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College Attendance Revisited:
Practice Problems:
5. Practice finding probabilities for z and t. Use the tables or your calculator. Remember:
P(z<-1.07)
normalcdf(-10000,-1.07,0,1)
P(t with 14 df > 2.52)
tcdf(2.52,10000,14)
a. P(z>1.56)
b. P(z<-0.94)
c. P(z<-2.59 or x>2.59)
d. With n=10: P(t>2.33)
e. With n=10: P(t<-1.50)
f. With n=20: P(t<-2.45)
g. With n=20: P(t<-1.37 or t>1.37)
6. At Rochester Institute of Technology, 34% of the students are female. The Department of
Mathematics and Statistics would like to know if the Data Analysis course has a different
percentage of female students.
a. Determine the population type and describe the population characteristic in words.
b. State H0 and Ha.
c. Is this a z or t test statistic?
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d. A random sample of 36 Data Analysis students had 16 women. Calculate the test statistic and pvalue.
e. Based on your p-value, is data at least as inconsistent with H0 as our sample likely to occur when
H0 is true? Explain.
7. It is well known that higher vertebrates – mammals and birds – exhibit lateralized behaviors; in
humans this is referred to as “handedness.” An investigator recently observed the coiling
behavior of cottonmouth snakes. He created a “laterality index” that measured the tendency for
snakes to coil clockwise or counterclockwise. If the snakes failed to exhibit laterality they would
have a laterality index equal to 0.5. The investigator wishes to determine whether juvenile
cottonmouths exhibit handedness.
a. What is the appropriate null hypothesis in this study?
b. What is the appropriate alternative hypothesis in this study?
c.
In the context of this study, describe a Type I and a Type II error
Suggested Homework: Read 10.3; Problems 25-27, 38, 69, 70
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10.4: Hypothesis Tests for a Population Mean
Assumptions for a Confidence Interval for a Population Mean:
1.
2.
3.
One-Sample t-test for a Population Mean
Null Hypothesis:
Test Statistic:
Alternative Hypothesis:
P-value:
Assumptions:
1.
2.
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A study conducted by researchers at Pennsylvania State University investigated whether time
perception, an indication of a person’s ability to concentrate, is impaired during nicotine withdrawal.
After a 24-hour smoking abstinence, 20 smokers were asked to estimate how much time had elapsed
during a 45-second period. Researchers wanted to see whether smoking abstinence had a negative
impact on time perception, causing elapsed time to be overestimated. Suppose the resulting data on
perceived elapsed time (in seconds) were as follows:
69
65
72
73
59
55
39
52
67
57
56
50
70
47
56
45
70
64
67
53
Mean and Standard Deviation:
State the hypotheses:
Assumptions:
Statistic and P-value
Conclusion:
Confidence Interval:
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A growing concern of employers is time spent in activities like surfing the Internet and emailing
friends during work hours. The San Luis Obispo Tribune summarized the findings of a large survey
of workers in an article that ran under the headline “Who Goofs Off More than 2 Hours a Day? Most
Workers, Survey Says” (August 3, 2006). Suppose that the CEO of a large company wants to
determine whether the average amount of wasted time during an 8-hour day for employees of her
company is less than the reported 120 minutes. Each person in a random sample of 10 employees
was contrasted and asked about daily wasted time at work. The resulting data are the following:
108
112
117
130
111
131
113
113
105
128
Mean and Standard deviation:
State the hypotheses:
Assumptions:
Statistic and P-value
Conclusion:
Possible Error:
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Practice Problems:
8.
In cities and towns on the borders between states there is “flight” across state lines to avoid
high state taxes on gasoline. Some states have large rivers for borders and tolls to cross bridges. Do
these tolls impede traffic to other states to buy cheaper gasoline? To test this hypothesis, an
experimental Toll-Free Week will be instituted at the Farmington Bridge in Iowa, where currently
approximately 50 cars per day drive back and forth. Let m denote the true average number of
border crossings per day at Farmington if there were no toll.
a. What is the appropriate null hypothesis in this study?
b. What is the appropriate alternative hypothesis in this study?
c. In the context of this study, describe a Type I and a Type II error.
9.
The demographics of television viewers are important factors in selling advertising time. The
RX Pharmaceutical company would like to market a new acid-reflux medication to consumers under
the age of 50. They are considering buying advertising time on the cable channel MSNBC, if they find
evidence that the average ago of MSNBC viewers is under 50 years.
a. Determine the population type and describe the population characteristic in words.
b. State H0 and Ha.
c. Is this a z or a t test statistic?
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d. Suppose that a random sample of 60 MSNBC viewers had a test statistic value of -1.807. Compute
the p-value.
e. Based on your p-value, is data at least as inconsistent with H0 as our sample likely to occur when
H0 is true? Explain.
10.
A recent article in Chance Magazine states that “For every age, all the way through the mid90s, male driving fatalities are typically 3 to 5 times that of female fatalities.” In other words, at least
75% of driving fatalities are male. The data for the article indicates that, in 2003, 414 male and 120
female 20-year-old drivers were killed while traveling alone. Consider this data to be a random
sample of all fatal crashes for 20-year-old drivers traveling alone. Does the data provide sufficient
evidence to conclude that more than 75% of all fatal crashes for 20-year-old drivers traveling alone
involve male drivers?
a. Determine the population type, describe the population characteristic in words and state the
hypotheses.
b. Set a reasonable value for α and write a formula of the test statistic.
c. Describe the sample (n, 𝑥̅ , 𝑝̂ , s?) and check that the sample meets the necessary assumptions.
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d. Compute the value of the test statistic and compute the p-value using the 1-PropZTest on your
calculator.
e. Reject or Fail to Reject H0 and make a concluding statement.
11.
Research indicates that 10% of all people are left-handed. A study of 1650 people age 65 and
older contained only 83 lefties. Does this data provide evidence that the proportion of elderly people
who are left handed smaller than the proportion in the general population?
f. Determine the population type, describe the population characteristic in words and state the
hypotheses.
g. Set a reasonable value for α and write a formula of the test statistic.
h. Describe the sample (n, 𝑥̅ , 𝑝̂ , s?) and check that the sample meets the necessary assumptions.
i.
Compute the value of the test statistic and compute the p-value.
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j.
Reject or Fail to Reject H0 and make a concluding statement.
12.
A nutritionist claims that ready-to-eat breakfast cereal has about 100 calories per ounce, on
average. A random sample of twelve ready-to-eat cereals provided the following nutritional
information:
Cereal Name
Calories Per Serving
Serving Size
Calories per
(grams)
ounce
Raisin Bran
190
59
91.14
Cocoa Krispies
120
31
Corn Flakes
100
28
Honey Bunches of Oats
130
32
Shredded Wheat
170
49
Honey Comb
120
32
Life
120
32
Puffed Rice
50
14
Cheerios
110
30
Lucky Charms
110
27
Wheaties
100
27
Wheat Chex
160
47
a.
Using the conversion factor of 1 ounce=28.3 grams, calculate the calories per ounce for each
cereal.
b.
Determine if the sample provides sufficient evidence to contradict the nutritionist’s claims.
Make sure you show all necessary work, label all values, check any necessary conditions and write a
complete sentence as your conclusion.
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13.
A boat manufacturer claims that a particular boat and motor combination will burn less than
4.0 gallons of fuel per hour. Fuel consumption for a random sample of 10 similar boats resulted in
the data below:
4.06
4.29
4.26
4.64
4.23
3.93
3.64
4.13
3.93
3.86
Is there sufficient evidence to conclude that the manufacturer's claim is correct? Use a = .05 and
test the appropriate hypothesis.
14.
Standard bracelet size is 7 inches (17.8 cm). Using our class as a sample, determine if that size
accommodates the average adult wrist size. Use the T-Test on your calculator.
Suggested Homework: Read 10.4; Problems 42, 44, 47, 48, 52, 55-57
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10.5: Power and the Probability of Type II Error
𝐻0 is true
𝐻0 is false
Reject 𝐻0
Fail to reject 𝐻0
Suppose that the student body president at a university is interested in studying the amount of
money that students spend on textbooks each semester. The director of financial aid services believes
that average amount spent on textbooks is $500 each semester, and uses this to determine the
amount of financial aid for which a student is eligible. The student body president plans to ask each
student in a random sample how much he or she spent on books this semester and use the data to
test (using a = .05) the following hypotheses:
H0: µ= 500 versus Ha: µ > 500
The power of the test depends on the true value of the mean! Because the actual value of µ is
unknown, we cannot know the power for the actual value of µ.
a
Textbooks…
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What is the probability of committing a Type I error?
If µ=500 is true, for what values of the sample mean would you reject the null hypothesis?
What is the probability of a Type II error?
What is the power of the test if µ=520?
What if µ=530?
Notice that, as the distance between the null hypothesized value for µ and our alternative value for µ
increases,β decreases AND power increases.
What if µ=510?
What happens if we use α=.01?
What happens to α, β and power when the sample size is increased?
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Power is the probability of rejecting the null hypothesis when in fact it is false.
Effects of Various Factors on the Power of a Test
1.
2.
3.
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Practice Problems:
15. Other things being equal, which of the following actions will reduce the power of a hypothesis
test?
I. Increasing sample size.
II. Increasing significance level.
III. Increasing beta, the probability of a Type II error.
(A) I only
(B) II only
(C) III only
(D) All of the above
(E) None of the above
16.
A package delivery company claims that it is on time 90% of the time. Some of its clients aren't
so sure, thinking that there are often delays in delivery beyond the time promised. The company
states that it will change its delivery procedures if it are wrong in its claim. Suppose that, in fact,
there are more delays than claimed by the company. Which of the following is equivalent to the power
of the test?
A. The probability that the company will not change its delivery procedures
B. The P-value > α
C. The probability that the clients are wrong
D. The probability that the company will change its delivery procedures
E. The probability that the company will fail to reject H0
Suggested Homework: Read 10.5; Problems 60, 61
Extra Credit: 10.6: 66, 67, 72, 74, 77, 80
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