1. 1
Week 8:
● Hypothesis Testing
with One-Sample t-test

2. 2
One Sample t-Test
 The shortcoming of the z test is that
it requires more information than is
usually available.
 To do a z-test, we need to know the
value of population standard
deviation to be able to compute
standard error. But it is rarely known.

3. 3
When the population variance is
unknown, we use one sample t-test
n
s
x 

What if σ is unknown?
•Can’t compute z test statistics (z score)
Z =
Population standard
deviation must be known
• Can compute t statistic
t =
n
x



Sample standard deviation
must be known

4. 4
Z-test or t-test?
 Do children with low self-esteem show
significantly more depression than
children in general? The average
depression score for the general
population is 90, with a deviation of 14.
 Do children with low-self esteem take on
a leadership role in a group project
significantly less than two times a
semester?
 Is the average GPA of freshman
admitted to OSU significantly higher or
lower than 3.0 in 2007?

5. 5
Hypothesis testing with a
one-sample t-test
 State the hypotheses
 Ho: μ = hypothesized value
 H1: μ ≠ hypothesized value
 Set the criteria for rejecting Ho
 Alpha level
 Critical t value

6. 6
Determining the criteria for
rejecting the Ho
 The t- value is used just like a z-statistic: if the
value of t exceeds some threshold or critical
valued, t , then an effect is detected (i.e. the null
hypothesis of no difference is rejected)

7. 7
Table C.3 (p. 638 in text)

8. 8
Degrees of freedom for One
Sample t-test
 Degrees freedom (d.f.) is computed
as the one less than the sample size
(the denominator of the standard
deviation):
df = n - 1

9. 9
Finding Critical Values
The t-distribution for df = 3, 2-tailed α = 0.10

10. 10
Finding Critical Values
The t-distribution for df =15, 2-tailed α = 0.05

11. 11
Finding Critical Values
The t-distribution for df =15, one-tailed α = 0.05

12. 12
Hypothesis testing with a
one-sample t-test
 Compute the test statistic (t-statistic)
t =
 Make statistical decision and draw
conclusion
 t ≥ t critical value, reject null hypothesis
 t < t critical value, fail to reject null
hypothesis
n
s
x 


13. 13
Decision about Ho

14. 14
One Sample t-test Example
 You are conducting an experiment
to see if a given therapy works to
reduce test anxiety in a sample of
college students. A standard measure
of test anxiety is known to produce a µ
= 20. In the sample you draw of 81
the mean = 18 with s = 9.
 Use an alpha level of .05

15. 15
Write hypotheses
 Ho: The average test anxiety in the
sample of college students will not be
statistically significantly different than
20.
 Ho: μ = 20
 H1 = The average test anxiety in the
sample of college students will be
statistically significantly lower than 20.
 H1: μ < 20

16. 16
Set Criterion
 αone-tailed = .05
 df = n – 1
 81 – 1 = 80
 t critical value = - 1.671
 Use closest and most conservative
value if exact value not given

17. 17
Compute test statistic (t
statistic)
t = 18 – 20 = -2 = -2
9 / √81 1

18. 18
Compare to criteria and
make decision
 t-statistic of -2 exceeds your critical
value of -1.671.
 Reject the null hypothesis and
conclude that average test anxiety in
the sample of college students is
statistically significantly lower than 20,
t = -2.0, p < .05.