Hypothesis testing
For one sample (Z and t-tests)
For two samples (Z and t-tests)
1st situation: When population standard deviation “σ” is known (Z-test)
1. A researcher claims that the average cost of men’s athletic shoes is less than $80. He
selects a random sample of 36 pairs of shoes from a catalog and finds the following
costs (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough
evidence to support the researcher’s claim at α = 0.10? Assume σ=19.2.
60, 70, 75, 55, 80, 55, 50, 40, 80, 70, 50, 95, 120, 90, 75, 85, 80, 60, 110, 65, 80, 85, 85, 45,
75, 60, 90, 90, 60, 95, 110, 85, 45, 90, 70, 70
2. The Medical Rehabilitation Education Foundation reports that the average cost of
rehabilitation for stroke victims is $24,672. To see if the average cost of rehabilitation
is different at a particular hospital, a researcher selects a random sample of 35 stroke
victims at the hospital and finds that the average cost of their rehabilitation is $26,343.
The standard deviation of the population is $3251. At α = 0.01, can it be concluded
that the average cost of stroke rehabilitation at a particular hospital is different from
$24,672?
3. A researcher wishes to test the claim that the average cost of tuition and fees at a four
Year public college is at most $5700. She selects a random sample of 36 four-year
public colleges and finds the mean to be $5950. The population standard deviation is
$659. Is there evidence to support the claim at α=0.05?
4. The average depth of the Hudson Bay is 305 feet. Climatologists were interested in
seeing if the effects of warming and ice melt were affecting the water level. Fifty-five
measurements over a period of weeks yielded a sample mean of 306.2 feet. The
population variance is known to be 3.57. Can it be concluded at the 0.05 level of
significance that the average depth has increased?
5. The average 1-year-old (both genders) is 29 inches tall. A random sample of 30 1-
year-olds in a large day care franchise resulted in the following heights. At α = 0.05,
can it be concluded that the average height differs from 29 inches? Assume σ=2.61.
25, 32, 35, 25, 30, 26.5, 26, 25.5, 29.5, 32, 30, 28.5, 30, 32, 28, 31.5, 29, 29.5, 30, 34,
29, 32, 27, 28, 33, 28, 27, 32, 29, 29.5
2nd case: When population standard deviation “σ” is unknown and sample size
“n” is also small i.e. less than 30 (t-test)
1. A medical investigation claims that the average number of infections per week at
a hospital in southwestern Pennsylvania is 16.3. A random sample of 10 weeks
had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is
there enough evidence to reject the investigator’s claim at α=0.05?
2. According to the American Pet Products Manufacturers Association, cat owners
spend an average of at least $179 annually in routine veterinary visits. A random
sample of local cat owners revealed that 10 randomly selected owners spent an
average of $205 with s = $26. Is there a significance statistical difference at α =
0.01?
3. A state executive claims that the average number of acres in western Pennsylvania
state parks is less than 2000 acres. A random sample of five parks is selected, and
the number of acres is shown. At α = 0.01, is there enough evidence to support the
claim?
959, 1187, 493, 6249, 521
“Two samples” (Independent case)
(Samples must be independent of each other. That is, there can be no
relationship between the subjects in each sample.)
3rd case: z-test for difference of means, when population standard deviations σ1 and
σ2 are known.
1. In a study of women science majors, the following data were obtained on two
groups, those who left their profession within a few months after graduation
(leavers) and those who remained in their profession after they graduated
(stayers). Test the claim that those who stayed had a higher science grade
point average than those who left. Use α=0.05.
Leavers Stayers
x 1 = 3.16 x 2 = 3.28
σ1 = 0.52 σ2 = 0.46
n1 = 103 n2 = 225
2. In a study, the researchers collected the data shown here on a self-esteem
Questionnaire. At α = 0.05, can it be concluded that there is a difference in the
self-esteem scores of the two groups:
Group 1 Group 2
x 1 = 3.05 x 1 = 2.96
σ1 = 0.75 σ2 = 0.75
n1 = 103 n2 = 225
3. The dean of students wants to see whether there is a significant difference in
ages of resident students and commuting students. She selects a sample of 50
students from each group. The ages are shown here. At α = 0.05, decide if
there is enough evidence to reject the claim of no difference in the ages of the
two groups.
Resident students
22, 25, 27, 23, 26, 28, 26, 24, 25. 20, 26, 24, 27, 26, 18, 19, 18, 30, 26, 18, 18, 19, 32, 23, 19, 19, 18,
29, 19, 22, 18, 22, 26, 19, 19, 21, 23, 18, 20, 18, 22, 21, 19, 21, 21, 22, 18, 20, 19, 23
Commuter students
18, 20, 19, 18, 22, 25, 24, 35, 23, 18, 23, 22, 28, 25, 20, 24, 26, 30, 22, 22, 22, 21, 18, 20, 19, 26, 35,
19, 19, 18, 19, 32, 29, 23, 21, 19, 36, 27, 27, 20, 20, 21, 18, 19, 23, 20, 19, 19, 20,25
4th case: t-test for difference of means (independent samples). When
population standard deviation is unknown and sample size is also less than
30.
1. The weights in ounces of a sample of running shoes for men and women are shown. Test
the claim that the mean of men shoes is greater than women.
Men Women
10.4, 12.6, 11.1, 14.7, 10.8, 12.9, 11.7, 10.6, 10.2, 8.8, 9.6, 9.5, 9.5, 10.1, 11.2,
13.3, 12.8, 14.5 9.3, 9.4, 10.3, 9.5, 9.8, 10.3, 11.0
2. The mean age of a sample of 25 people who were playing the slot machines is 48.7 years,
and the standard deviation is 6.8 years. The mean age of a sample of 35 people who were
playing roulette is 55.3 with a standard deviation of 3.2 years. Can it be concluded at α =
0.05 that the mean age of those playing the slot machines is less than those splaying
roulette?
3. The number of grams of carbohydrates contained in 1-ounce servings of randomly
selected chocolate and non-chocolate candy is listed here. Is there sufficient evidence to
conclude that the difference in the means is significant? Use level of significance of 1%.
Chocolate Non-chocolate
29, 25, 17, 36, 41, 25, 32, 29, 38, 41, 41, 37, 29, 30, 38, 39, 10, 29,
34, 24, 27, 29 55, 29
5th case: t-test for difference of means (dependent samples).
1. A dietitian wishes to see if a person’s cholesterol level will change if the diet is
supplemented by a certain mineral. Six subjects were pretested, and then they took the
mineral supplement for a 6-week period. The results are shown in the table.
(Cholesterol level is measured in milligrams per deciliter.) Can it be concluded that
the cholesterol level has been changed at α = 0.10?
Subject 1 2 3 4 5 6
Before (x1) 210 235 208 190 172 244
After (x2) 190 170 210 188 173 228
2. An obstacle course was set up on a campus, and 10 volunteers were given a chance to
complete it while they were being timed. They then sampled a new energy drink and
were given the opportunity to run the course again. The “before” and “after” times in
seconds are shown below. Is there sufficient evidence at α = 0.05 to conclude that the
students did better the second time?
Student 1 2 3 4 5 6 7 8
Before 67 72 80 70 78 82 69
75
After 68 70 76 65 75 78 65
68
3. A veterinary nutritionist developed a diet for overweight dogs. The total volume of
food consumed remains the same, but one half of the dog food is replaced with a low-
calorie “filler” such as canned green beans. Six overweight dogs were randomly
selected from her practice and were put on this program. Their initial weights were
recorded, and then they were weighed again after 4 weeks. At the 0.05 level of
significance can it be concluded that the dogs lost weight?
Before 42 53 48 65 40 52
After 39 45 40 58 42 47
