HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2008 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Section 10.2
Hypothesis Testing for Means
(Small Samples)
What this lesson is about
• Learn to perform a hypothesis test
• The previous lesson was only about how to set
up a hypothesis test.
– Reading and understanding the real-life scenario.
– Getting the right letter, μ or p.
– Getting the right relational operators in the right
places: = and ≠, ≤ and >, ≥ and <.
– Getting the right value of μ or p (and setting aside
the “noise” numbers in the problem statement.)
(Added content by D.R.S.)
Choice: Do a t Test or a z Test?
Small Samples: t Test
• “Small” means “sample size
is n < 30.
• There’s an assumption that
the population is normally
distributed.
• If the population is not
normally distributed, this
method we use is NOT valid.
• Easy for today: everything
we do is a t Test.
Large Samples: z Test
• “Large” means “sample size
is n ≥ 30.
• To be discussed in a later
lesson.
• The Bluman book has
slightly different rules from
the way this Hawkes book
does it. Just be aware of
that.
(Added content by D.R.S.)
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Test Statistic for Small Samples, n < 30:
with d.f. = n – 1
To determine if the test statistic calculated from the
sample is statistically significant we will need to look at
the critical value.
The critical values for n < 30 are found from the
t-distribution.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Find the critical value:
Find the critical t-score for a right-tailed test that has 14
degrees of freedom at the 0.025 level of significance.
Solution:
d.f. = 14 and a = 0.025
t0.025 = 2.145
(Added info)
• It’s in Table C, Critical Values
of t
Inputs:
• Column for α (alpha)
• Choose the right column for
one- or two-tailed
• Row for d.f., degrees of
freedom (= sample size n,
minus 1)
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Rejection Regions:
Determined by two things:
1. The type of hypothesis test.
2. The level of significance, a.
Finding a Rejection Region:
1. Look up the critical value, tc, to determine the
cutoff for the rejection region.
2. If the test statistic you calculate from the sample
data falls in the a area, then reject H0.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Types of Hypothesis Tests:
Alternative Hypothesis
Type of Test
< Value
> Value
≠ Value
Left-tailed test
Right-tailed test
Two-tailed test
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Rejection Regions for Left-Tailed Tests, Ha contains <:
Reject if t ≤ –ta
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Rejection Regions for Right-Tailed Tests, Ha contains >:
Reject if t ≥ ta
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Rejection Regions for Two-Tailed Tests, Ha contains ≠:
Reject if | t | ≥ ta/2
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Steps for Hypothesis Testing:
IMPORTANT !!!!
1. State the null and alternative hypotheses.
2. Set up the hypothesis test by choosing the
test statistic [that is, make a decision on
whether it’s a t or z problem] and determining
the values of the test statistic that would lead
to rejecting the null hypothesis [the critical
value(s)].
3. Gather data and calculate the necessary
sample statistics [t = or z = ].
4. Draw a conclusion [Stating it two ways:
reject/fail to reject, and also in plain English].
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Draw a conclusion:
27 college students are asked how many parking tickets they
get a semester to test the claim that the average student gets
more than 9 tickets a semester. The sample mean for the
number of tickets was found to be 9.8 with a standard
deviation of 1.5. Use a = 0.10.
Solution:
H0: μ ≤ 9 tickets Ha: μ > 9 tickets.
n = 27,  = 9,
= 9.8, s = 1.5, d.f. = 26, a = 0.10
This is the CRITICAL VALUE. Either use table or
t0.10 = 1.315 invT(0.10,26). Draw a PICTURE, too. Mark 1.315
and highlight the critical region.
2.771 This is the TEST STATISTIC.
Mark 2.771 on your picture.
Since t is greater than ta , we will reject the null hypothesis.
Remarks about the parking ticket
example
• There was a choice made to do a t Test
because the sample size was < 30.
• There was an implicit assumption that the
distribution of the count of parking tickets fits
a normal distribution.
• It was a RIGHT-TAILED TEST because of the “>”
in the alternative hypothesis.
(Added content by D.R.S.)
Remarks about the parking ticket
example, continued
• Hypothesis tests are really essay questions.
• The outline for the essay is the four-step
procedure described in the earlier slide.
• Each of the four steps needs to be explained
plainly with a lot of words: Complete
thoughts and complete sentences.
• The final statement is in plain English, suitable
for the general public to understand.
(Added content by D.R.S.)
The Parking Ticket problem done
as an essay question
1. State the hypotheses
• We investigate the claim
that the average student
receives more than nine
parking tickets in a
semester. Our hypotheses
are:
• Null hypothesis, H0: μ ≤ 9
• Alternative hypothesis:
Ha: μ > 9, more than nine
tickets per semester.
2. Find the critical value
• This is a t Test, right tailed.
• The sample size is n = 27.
• The degrees of freedom is
d.f. = n – 1 = 26.
• The level of significance
chosen is α = 0.10
• The critical value is
tα=0.10,d.f.=26 = 1.315
(Added content by D.R.S.)
The Parking Ticket problem done
as an essay question
3. Compute the test statistic
• (As shown on the earlier
slide – formula & details)
2.771
4. Conclusions
• Since the test value 2.771 is
greater than the critical
value 1.315, we reject the
null hypothesis.
• “There is sufficient evidence
to support the claim that
the average student gets
more than 9 parking tickets
per semester.”
(Added content by D.R.S.)
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Draw a conclusion:
A hometown department store has chosen its marketing strategies for many
years under the assumption that the average shopper spends no more than
$100 in their store. A newly hired store manager claims that the current
average is higher, and wants to change their marketing scheme accordingly.
A group of 24 shoppers is chosen at random and found to have spent on
average $104.93 with a standard deviation of $9.07. Test the store manager’s
claim at the 0.010 level of significance.
Solution:
First state the hypotheses:
H0:  ≤ 100
Ha:  > 100
Next, set up the hypothesis test and determine the critical
value:
d.f. = 23, a = 0.010
t0.010 = 2.500
Reject if t ≥ ta , or if t > 2.500.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Solution (continued):
Gather the data and calculate the necessary sample statistics:
n = 24,  = 100, = 104.93, s = 9.07,
2.663
Finally, draw a conclusion:
Since t is greater than ta , we will reject the null hypothesis.
Added content
• Repeating several of the slides with extra
comments about TI-84
• Also an important reminder: using this
method for small sample sizes requires that
the population being studied is NORMALLY
DISTRIBUTED. Not uniform, not skewed, but
a bell curve distribution is assumed. (This
book somewhat glosses over this point.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
The critical values for n < 30 are found from the
t-distribution.
Find the Find the critical t-score for a right-tailed test that has 14
critical
degrees of freedom at the 0.025 level of significance.
value:
d.f. = 14 and a = 0.025
Solution: t
0.025 = 2.145
invT(area to left, d.f.) = t value
Plus or Minus Sign? Either by
symmetry or by adjusting the
area value for a right-tailed test.
You still have to understand
whether it’s left-tailed, righttailed, or two-tailed. The
calculator won’t do that for you !
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Draw a conclusion:
27 college students are asked how many parking tickets they
get a semester to test the claim that the average student gets
more than 9 tickets a semester. The sample mean for the
number of tickets was found to be 9.8 with a standard
deviation of 1.5. Use a = 0.10.
Solution:
n = 27,  = 9,
t0.10 = 1.315
= 9.8, s = 1.5, d.f. = 26, a = 0.10
Again, fix up the sign by
knowing that it’s a right-tailed
test, therefore positive critical
value. The calculator will not
do this thinking for you.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
(continued from previous slide)
Solution:
n = 27,  = 9,
= 9.8, s = 1.5, d.f. = 26, a = 0.10
EXTRA ( ) around complicated
numerators and denominators !!!
t0.10 = 1.315
2.771
Since t is greater than ta ,
we will reject
the null hypothesis.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Draw a conclusion:
A hometown department store has chosen its marketing strategies for many
years under the assumption that the average shopper spends no more than
$100 in their store. A newly hired store manager claims that the current
average is higher, and wants to change their marketing scheme accordingly.
A group of 24 shoppers is chosen at random and found to have spent on
average $104.93 with a standard deviation of $9.07. Test the store manager’s
claim at the 0.010 level of significance.
Solution:
First state the hypotheses:
H0:  ≤ 100
Ha:  > 100
Next, set up the hypothesis test
and determine the critical value:
d.f. = 23, a = 0.010
t0.010 = 2.500
Reject if t ≥ ta , or if t > 2.500.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Solution (continued):
Gather the data and calculate the necessary sample statistics:
n = 24,  = 100, = 104.93, s = 9.07,
2.663
Finally, draw a conclusion:
Since t is greater than ta , we will reject the null hypothesis.
TI-84 T-Test
• The TI-84 has a built in Hypothesis Testing tool
• STAT menu, TESTS submenu, 2:T-Test
• You must understand how to do hypothesis
testing with charts and formulas, however.
The calculator is not a substitute for that.
Mere button smashing will lead you to failure.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Example:
27 college students are asked how many parking tickets they
get a semester to test the claim that the average student gets
more than 9 tickets a semester. The sample mean for the
number of tickets was found to be 9.8 with a standard
deviation of 1.5. Use a = 0.10.
Solution:
Sample’s
Mean,
Standard
deviation,
and Size
Highlight “Calculate” and press ENTER
Choose “Data” if the 27 data
values were in TI-84 Lists,
Stats if we have summary
statistics already calculated
Null hypothesis’s mean
Direction of the
Alternative Hypothesis
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
Example, continued:
. . . . Use a = 0.10.
Verify that it did the
Test you wanted and that it
has the correct Alternative
Hypothesis. Verify that
the sample data is correct.
When using the calculator’s
T-Test, we use the “p-value
method”. You don’t need a t
critical value. Instead, you
compare your p-value to the
α (alpha) level of
significance.
If your p < α(alpha), then
the decision is “Reject H0”.
The t= is the Test Statistic. It comes from the
same formula as the one we’ve been using.
The p = is the p-value. It is the area to the
right of that t value (in the case of this righttailed test.) It is the probability of getting a t
value as extreme as the t value we got.
HAWKES LEARNING SYSTEMS
Hypothesis Testing
math courseware specialists
10.2 Hypothesis Testing for Means
(Small Samples)
The other example, done with TI-84 T-Test and the p-value method:
A hometown department store has chosen its marketing strategies for many
years under the assumption that the average shopper spends no more than
$100 in their store. A newly hired store manager claims that the current
average is higher, and wants to change their marketing scheme accordingly.
A group of 24 shoppers is chosen at random and found to have spent on
average $104.93 with a standard deviation of $9.07. Test the store manager’s
claim at the 0.010 level of significance.
H0:  ≤ 100
Ha:  > 100
Compare your p-value p=.0069501788 to alpha: α=0.010
and make the decision: Should we reject H0?