MAT 170 – Confidence Interval and Hypothesis Test Review Problems
1) A manufacturer states that the average lifetime of its light bulbs is 3 years or 36 months
and the standard deviation is 8 months. Fifty bulbs are selected, and the average lifetime
is found to be 32 months. Should the manufacturer’s statement be rejected at the 0.01
significance level?
a) Identify if the problem involves a t-test or a z-test.
b) State the null hypothesis as a sentence. Ho:   36
c) Perform a hypothesis test using your calculator, and write down your results for z/t
and p. z=-3.54, p-value < .01
d) Explain what the p-value means in a sentence. There is less than a 1% chance of
observing sample data as extreme as we did if in fact the data comes from a
population with mean 120.
e) Determine if you do or do not reject the null hypothesis. Reject the null hypothesis
since we did get the data (which was very unlikely to happen). There is strong
evidence to suggest that the mean lifetime of this brand of light bulbs is not 36
months.
2) The mean yield of corn in the United States is about 120 bushels per acre. A survey of
40 farmers this year gives a sample mean yield of 123.8 bushels per acre. We want to
know whether this is good evidence that the national mean this year is not 120 bushels
per acre. Assume that the farmers surveyed are an SRS from a population of all
commercial corn growers and that the standard deviation of the yield in this population
is 10 bushels per acre.
a) State the appropriate null hypothesis. . Ho:   120
b) Carry out the test. z=2.4, p-value =.02
c) Are your results significant at the 5% level? Yes At the 1% level? No
d) State your conclusions in words a non-statistician could understand. At the 1% level,
there is not enough evidence to suggest that the mean bushels per acre this year
differs from 120.
e) Based on the sample data, compute a 95% confidence interval for the national mean
yield of bushels of corn per acre. State your conclusions in words a non-statistician
could understand? (120.6,127)
f) Based on the sample data, compute a 99% confidence interval for the national mean
yield of bushels of corn per acre. State your conclusions in words a non-statistician
could understand. (119.52, 128.08)
g) Compare your results in e. and f. with your results in c. Notice that the 95% CI
does not contain   120 so we would reject the null hypothesis at the 5% level.
However, the 99% CI does contain   120 so we would not reject the null
hypothesis at the 1% level.
3) A national magazine claims that the amount of t.v. that the average college student
watches per week differs from the general public. The national average is 29.4 hours per
week, with a standard deviation of 2 hours. A sample of 25 college students has a mean
of 27 hours. Is the result significant at the 1% level? Ho:   29.4; z=-6 so p-value is
.0000000019. The results are extremely significant!
4) To study the birth weights of infants whose mothers smoke, a physician records the
weights of 100 newborns whose mothers smoke. She finds that the average weight is 6.1
pounds with a standard deviation of 2.1 pounds.
a) Find a 99% confidence interval that estimates the mean birth weight of children of
smoking mothers. (5.55, 6.65)
b) State your conclusions using language a non-statistician could understand. With 99%
confidence, the mean weight for all infants whose mothers smoke is between 5.55
and 6.65 pounds.
5) An admissions counselor states that the average cost for one year’s tuition for all colleges
in the state of Pennsylvania is $6250. A sample of 15 colleges is selected, and the
average tuition is found to be $6801. Suppose that it is known that the standard
deviation is $1075. Test the hypothesis, at the 1% level, to see if the average cost of
tuition is $6250. Ho:   6250; z=1.99; p-value =.047. At the 1% level there is not
enough evidence to suggest that the average cost of tuition for all colleges in PA differs
from $6250.
6) The accompanying values are measured maximum breadths of male Egyptian skulls
from different epochs. Changes in head shape over time suggest that interbreeding
occurred with immigrant populations. Use the Kruskal-Wallis test with a 0.05
significance level to test the claim that the three samples come from identical
populations. Include these steps:




Write the null hypothesis as a sentence. Ho:= The Egyptian skulls all come from
one population.
Calculate H, showing all your work H=6.63
Decide if you reject or do not reject .01<p-value<.05
Write a final sentence that begins as either “There was a significant difference…”
or “No significant difference was found …”
4000 BC
131
138
125
129
132
135
132
134
138
1850 BC
129
134
136
137
137
129
136
138
134
150 AD
128
138
136
139
141
142
137
145
137
7.Run an ANOVA test ( = 0.05) to compare income for various levels of education,
using this data (given in thousands of dollars):
high school:
college:
6
30
15
30
24
31
24
39
30
60
40
74
62
grad. school:
27
49
58
78
83
102
You may assume that salaries are normally distributed.
a.
Why is the ANOVA test appropriate? Be specific.1 categ vs. 1 numer. Var; data
come from one population which is normally distributed
b. Write the null hypothesis as a sentence. Ho: There is no correlation between level
of education and income levels.
c. Give F and p values from the calculator, rounded to four places after the decimal.
F=4.96; p-value =.02.
d.
The FACTOR df is __2___ and the ERROR df is ___16___.
e. What is your conclusion? (reject or do not reject). Reject. There is statistically
strong evidence to suggest that there is a correlation between education level and income.