MATH221
Integrated Learning System Power
Point Lecture Week 5
Hypothesis Testing
Hypothesis Testing for the
Mean: Small Samples
The only difference between hypothesis testing for the
population mean with small (n < 30) samples and
hypothesis testing for the mean with large (n  30)
samples, which we covered on-site, is the test statistic
used in the test. For small samples we will use the t
statistic.
The t-test for a Mean
t 
x 
s /
n
Example
A computer salesman believes that the mean repair cost
for damaged computers is more than $95. To test this
claim, you determine the repair cost for seven randomly
selected computers and find that the mean repair cost is
$100 per computer with a standard deviation of $42.50.
At  = 0.01, do you have enough evidence to support
the repairman’s claim?
Solution
First we state the Hypothesis.
Ho:   $95
Ha:  > $95 (claim)
n = 7, so the degrees of freedom are 6.
Solution (continued)
This is a right-tailed test so looking up the critical value
for t with  = 0.01 and d.f. = 6 we get t0.01,6 = 3.143. So
we will reject Ho if t > 3.143.
t 
100  95
42 . 5 /
7
 0 . 311
So we fail to reject Ho and conclude that there is not
enough evidence to support the repairman’s claim.
Using Technology: TI 83
T-Test
Inp t:D ata Stats
 o :95
x:100
Sx:42.5
n:7
 :  o <  o >  o
C alculate D raw
Using Technology: TI 83
(Cont.)
T-Test
>95
t=.3112648601
p=.3830651297
x=100
Sx=42.5
n=7
Student Exercise
An environmentalist estimates that the mean waste
recycled by American adults is more than one pound per
person per day. You want to test this claim. You find
that the mean waste recycled per person per day for a
random sample of 12 American adults is 1.2 pounds and
the standard deviation is 0.3 pounds. At  = 0.05, can
you support this claim?
Hypothesis Testing for
Proportions
Earlier we learned that if np > 5 and nq > 5 then the
sampling distribution for the sample proportions is normal
with:
 pˆ  p and  pˆ 
pq / n
The test statistic for the hypothesis test of proportions is:
z 
pˆ  p
pq / n
Example
A medical researcher estimates that no more than 55% of
American adults eat breakfast everyday. In a random
sample of 250 adults, 56.4% say that they eat breakfast
everyday. At  = 0.01, is their enough evidence to reject
the researcher’s claim?
Solution
Ho: p  0.5 (claim)
Ha: p > 0.5
Right tailed test so z0.01 = 2.326
Reject Ho if z > 2.326
z 
. 564  . 55
. 55 (. 45 ) / 250
 0 . 445
So we fail to reject Ho and conclude that there is not
enough evidence to reject the researcher’s claim.
Student Exercise
In your work for a business regulatory agency you find
that in a sample of 1762 Americans, 1004 of them
believe that government regulation of business does
more harm than good. At  = 0.02, can you reject the
claim that 60% of Americans have this view?
Hypothesis Testing for
Variance and Standard Dev.
If we want to manufacture items, say golf balls, that have
to be nearly exactly alike in all respects, we must be
concerned with controlling variation. If the population is
normal, you can test the variance and standard deviation
of the population using the chi-squared test with n – 1
degrees of freedom.
Guidelines for Chi-Square Test
1. Write the null and alternative hypotheses.
2. Specify .
3. The Critical values for the 2 distribution are found in
Table 6 of Appendix B in your text.
1. For a right-tailed test use the value corresponding
to  and the d.f.
2. For a left tailed test use the value corresponding to
1 -  and the d.f.
3. For a two-tailed test use the values corresponding
to the d.f. and /2, and 1 - /2.
2 Test Statistic

2

n
 1 s

2
2
Example
A travel agency estimates that the standard deviation of
room rates in a certain city is no more than $25. You
work for a consumer advocacy group and are asked to
test this claim. You find that a random sample of 13
hotel room rates has a standard deviation of $27.50. At
 = 0.01, do you have enough evidence to reject the
agency’s claim?
Solution
Ho:   $25.00 (claim)
Ha:  > $25.00
d.f. = 12
This is a right-tailed test so we look up 2 for  = 0.01
and d.f. = 12. The critical value is 2 = 26.217
Reject Ho if 2 > 26.217
Solution (continued)
Test Statistic:

2

13
 1 27 . 7 
2
25
2
 14 . 732
So we fail to reject the null hypothesis and conclude that
there is not enough evidence to reject the claim.
Student Exercise
A hospital spokesperson claims that the standard
deviation of the waiting times experienced by patients in
its minor emergency department is no more than 0.5
minutes. You doubt the validity of this claim. If a random
sample of 25 waiting times has a standard deviation of 0.7
minutes, can you reject the spokesperson’s claim? Use an
 of 0.10.