Slides to accompany lecture in a statistics class

1. One Sample Hypothesis Tests
z-Test and t-Test

2. H0 : μ after treatment = μ before treatment
HA : μ after treatment  μ before treatment

3. The problem
• A researcher is testing the effectiveness of a new
herbal supplement that claims to improve physical
fitness. A sample of n = 25 college students is obtained
and each student takes the supplement daily for six
weeks. At the end of the 6-week period, each student
is given the Marine Physical Fitness Test and the
average score for the sample is M = 38.68. For the
general population of college students, the distribution
of scores on the Marine Physical Fitness test is normal
with a mean of μ = 35 and a standard deviation of
σ = 10. Do students taking the supplement have
significantly better fitness scores?
Use a two-tailed test with α = .05.

4. Gather information from the problem
in an organized way as you read.
• n = 25 college students
• M = 38.68
• μ = 35
•  = 10
• Compute the Standard Error
M = 10/25 = 10/5 = 2

5. Step 1: State the Hypotheses
• Write the Null and Alternate Hypotheses
as sentences

6. Step 1: State the Hypotheses
• College students who take the herbal supplement
will have the same average score on the Marine
Physical Fitness Test as the general population of
students, or 35 points.
• College students who take the herbal supplement
will have a different average Marine Physical
Fitness score than the general population of
students.
• Two-tailed: “different average...score”
• One-tailed: “higher average…” or “lower average”

7. Step 1: State the Hypotheses
• Write the Null and Alternate Hypotheses
as equations

8. State the Hypotheses as Equations
• H0: M – μ = 0 or H0: M = 35
• HA: M – μ  0 or HA: M  35
• For a one-tailed test, you would need < and >
signs that, taken together, cover all possibilities
• H0: M – μ  0
• HA: M – μ > 0

9. Step Two: Locate the critical region
Step Two: Locate the critical region on the graph to match the
requirements stated in the problem. Be sure to write in the Critical
Value of the test statistic, and shade in the critical region.

10. Step Two: Locate the critical region
Step Two: Locate the critical region on the graph to match the
requirements stated in the problem. Be sure to write in the Critical
Value of the test statistic, and shade in the critical region.

11. Step Three: Compute the z test
statistic for this problem.
•
•
•
Use the numbers you
collected or computed
while reading problem
• n = 25 students
• M = 38.68
• μ = 35
•  = 10
• Standard Error
M = 10/25 = 2

m
M
z


38.68  35
2
z 
1.84
3.68
z  
2

12. Step Four: Make a decision
Reject H0 if z is in the rejection region
(i.e., if z > 1.96 or z < -1.96)
Otherwise, retain H0
z = 1.84
X

13. Recipe for writing up test results
• Sandwich: narrative outside, statistics inside
• Opening sentence that states the experiment’s
question and hypothesis.
• Sentence(s) that include
the descriptive statistics
(means, standard deviations)
• Statement of your decision:
results are significant (reject H0) or not significant
(retain H0) including test statistic and alpha-level
• Closing sentence(s) that answer the question

14. Write up the results in a narrative
A researcher was interested to find out whether an herbal
supplement had an impact on physical fitness as measured by the
Marine Physical Fitness Test. The average level of physical fitness
(M = 38.68) of 25 college students who took the herbal supplement
in the study was tested. This mean was found to be
(choose the appropriate one of the following options)
SIGNIFICANTLY HIGHER / NOT SIGNIFICANTLY DIFFERENT / SIGNIFICANTLY LOWER
than the national college student mean of μ = 35
(put test results in parentheses) ( z = 1.84, p > .05) .
[Write closing narrative sentence to summarize the findings]
There is not enough evidence in this sample to conclude that taking
the herbal supplement has any effect on general fitness level; the
higher average fitness score for this small sample of students might
have due to chance.

15. WHEN WE DO NOT KNOW THE
POPULATION STANDARD DEVIATION:
T-TESTS

16. H0 : μ after treatment = μ before treatment
HA : μ after treatment  μ before treatment

17.  Use s2 to estimate σ2
 Estimated standard error:
s
2
n
or
s
n
sm 
 Estimated standard error is used as
estimate of the real standard error when
the value of σm is unknown.

18.  The t-statistic uses the estimated standard
error in place of σm
M

m s
t

 The t statistic is used to test hypotheses
about a hypothesized (unknown) or known
population mean μ when the value of σ
is unknown

20.  Family of distributions, one for each value of
degrees of freedom.
 Approximates the shape of the normal
distribution
• Flatter than the normal distribution
• More spread out than the normal distribution
• More variability (“fatter tails”) in t distribution
 Use t values instead of Normal Curve Values

21.  To compute the standard deviation
(variance) we computed the mean first.
• Only n-1 scores in a sample are independent
• Researchers call n-1 the degrees of freedom
 Degrees of freedom
• Noted as df
• df = n-1 for a one-sample test

22. 1. State the null and alternative hypotheses
2. Select an alpha level
Locate the critical region using the
t distribution and df
3. Calculate the t test statistic
4. Make a decision regarding H0

23. df = 24
-2.064 2.064

24. z-test
 One sample
 Population mean μ is
known or hypothesized
 Population standard
deviation σ is known
t-test
 One sample
 Population mean μ is
known or hypothesized
 Population standard
deviation σ is NOT known
 
n n
m
2
or
 

m
M
z


s
n
or
2
s
n
sm 
,  1

M
 df n
s
t
m


25.  Also: Cronk 6.2 and Atomic Learning videos

26.  Select variable
 Enter test value
 Click OK

28. Rejection region defined by
alpha is colored in.
In this test, p-value is less area than the rejection region, so
we would reject H0 because the probability of getting a
result this extreme or more is less than 

29. Computing by hand
 Find the limits of the
rejection region from a
table or calculator for t
 If the t-statistic you
compute is more extreme
(larger in magnitude), then
REJECT H0
 Otherwise, RETAIN H0
Computing with SPSS
 Use the alpha level (often
.05) for comparison.
 If the p-value that SPSS
computes is less than the
alpha, REJECT H0
 Otherwise, retain H0.

30. A researcher was interested to find out whether an herbal
supplement had an impact on physical fitness as measured by the
Marine Physical Fitness Test. The average level of physical fitness
(M = 38.68, s = 9.017) of 25 college students who took the herbal
supplement in the study was tested. This mean was found to be
(choose the appropriate one of the following options)
SIGNIFICANTLY HIGHER / NOT SIGNIFICANTLY DIFFERENT / SIGNIFICANTLY LOWER
than the national college student mean of μ = 35
(put test results in parentheses) ( t = 2.041, p = .052) .
[Write closing narrative sentence to summarize the findings]
There is not enough evidence in this sample to conclude that taking
the herbal supplement has any effect on general fitness level; the
higher average fitness score for this small sample of students might
have due to chance.

31.  Try the analysis again with Sample 2.
 How are the results the same or different?
 How would we handle writing up the
different results?
[Hint: Cronk has two types of write-up in each
section of his book]