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Chapter 14
Tests of significance: the basics
BPS - 3rd Ed.
Chapter 14
1
Reasoning of Tests of Significance
What
would happen if we repeated
the sample or experiment many
times?
How likely would it be to see the
observed results if the claim was
untrue?
Do the data give evidence against a
claim?
BPS - 3rd Ed.
Chapter 14
2
Stating Hypotheses
Null Hypothesis, H0



The statement being tested is called the
null hypothesis
The null hypothesis is a statement of “no
effect” or “no difference”
The test is designed to assess the
strength of evidence against the null
hypothesis.
BPS - 3rd Ed.
Chapter 14
3
Stating Hypotheses
Alternative Hypothesis, Ha



The statement we are trying to find evidence for
is called the alternative hypothesis.
The alternative hypothesis usually indicates “an
effect” or “difference”
The alternative hypothesis expresses the
hopes or suspicions we bring to the data.
BPS - 3rd Ed.
Chapter 14
4
Case Study I: “Weight Gain”
Suppose we know that weight gain after the age of 30
varies from individual to individual according to a
Normal distribution with standard deviation s = 1 lbs
The symbol m represents the mean or expected weight
gain for all individuals (the parameter)
Ten individuals between the age of 30 and 40 yield an
average gains of 1.02 lbs. x
Do these data provide sufficient evidence that people
in this age range tend to gain weight each year?
BPS - 3rd Ed.

Chapter 14
5
Case Study I: “Weight Gain”



If the claim that m = 0 is true (no average weight gain),
the sampling distribution of x from 10 individuals is
Normal with m = 0 and standard deviation
σ
1

 0.316
n
10
The data yielded x = 1.02, which is more than three
 from m = 0. This provides strong
standard deviations
evidence that people gain weight.
If the data yielded x = 0.3, which is less than one
standard deviations from m = 0, there would be less
 evidence that individuals gains weight.
convincing
BPS - 3rd Ed.
Chapter 14
6
Case Study I
Weight gain
BPS - 3rd Ed.
Chapter 14
7
Statistical hypotheses
Null:
H 0: m = m 0
One
sided alternatives
Ha: m >m0
Ha: m <m0
Two sided alternative
Ha: m m0
BPS - 3rd Ed.
Chapter 14
8
Case Study I
Weight gain
The null hypothesis is “no average weight gain”
The alternative hypothesis “yes, average weight gain”
H0: m = 0
Ha: m > 0
We use a one-sided test because we are interested only
in determining weight gain (and not weight loss)
BPS - 3rd Ed.
Chapter 14
9
Test Statistic
Testing the Mean of a Normal Population
Take an SRS of size n from a Normal population
with unknown mean m and known standard
deviation s.
The test statistic for null hypothesis H0: m = m0 is
x  μ0
z
σ
n
BPS - 3rd Ed.
Chapter 14
10
Case Study I
Weight Gain
The test statistic for no average weight gain is:
x  μ0
1.02  0
z

 3.23
σ
1
10
n
This shows that the sample mean is more than 3
standard deviations above the hypothesized mean of 0.
This provides strong evidence against H0.
BPS - 3rd Ed.
Chapter 14
11
P-value
The P-value provides the probability that the
test statistic would take a value as extreme or
more extreme than the value observed if the
null hypothesis were true.
The smaller the P-value, the stronger the
evidence the data provide against the null
hypothesis
BPS - 3rd Ed.
Chapter 14
12
P-value for Testing Means

Ha: m> m0


Ha: m< m0


P-value is the probability of getting a value as large or
larger than the observed test statistic (z) value.
P-value is the probability of getting a value as small or
smaller than the observed test statistic (z) value.
Ha: mm0

P-value is two times the probability of getting a value as
large or larger than the absolute value of the observed test
statistic (z) value.
BPS - 3rd Ed.
Chapter 14
13
Case Study I
Weight Gain
For test statistic z = 3.23 and alternative hypothesis
Ha: m > 0, the P-value is:
P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006
Interpretations: If H0 is true, there is only a 0.0006
(0.06%) chance that we would see results at least as
extreme as those in the sample  we therefore have
evidence against H0 and in favor of Ha.
BPS - 3rd Ed.
Chapter 14
14
Case Study I
Weight gain
BPS - 3rd Ed.
Chapter 14
15
Statistical Significance



If the P-value is as small or smaller than the
significance level a (i.e., P-value ≤ a), then we say
that data are statistically significant at level a.
If we choose a = 0.05, we are requiring that the data
give evidence against H0 so strong that it would occur
no more than 5% of the time when H0 is true.
If we choose a = 0.01, we are insisting on stronger
evidence against H0, evidence so strong that it would
occur only 1% of the time when H0 is true.
BPS - 3rd Ed.
Chapter 14
16
Tests for a Population Mean
The four steps in carrying out a significance test:
1. State the null and alternative hypotheses.
2. Calculate the test statistic.
3. Find the P-value.
4. State your conclusion.
The procedure for Steps 2 and 3 is on the next page.
BPS - 3rd Ed.
Chapter 14
17
BPS - 3rd Ed.
Chapter 14
18
Case Study I
Weight gain problem
1.
2.
Hypotheses:
Test Statistic:
H 0: m = 0
H a: m > 0
z
x  μ0
σ

1.02  0
1
n
3.
4.
 3.23
10
P-value: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006
Conclusion:
Since the P-value is smaller than a = 0.01, there is strong
evidence that people gain weight in this age range.
BPS - 3rd Ed.
Chapter 14
19
Case Study II
Studying Job Satisfaction
Does the job satisfaction of assembly workers
differ when their work is machine-paced rather
than self-paced? A matched pairs study was
performed on a sample of workers, and each
worker’s satisfaction was assessed after
working in each setting. The response variable
is the difference in satisfaction scores, selfpaced minus machine-paced.
BPS - 3rd Ed.
Chapter 14
20
Case Study II
Studying Job Satisfaction
The null hypothesis “no average difference” in the
population of assembly workers. The alternative
hypothesis (that which we want to show is likely to be
true) is “there is an average difference in scores” in the
population of assembly workers.
H0: m = 0
Ha: m ≠ 0
This is considered a two-sided test because we are
interested determining a difference in either direction.
BPS - 3rd Ed.
Chapter 14
21
Case Study II
Studying Job Satisfaction
Suppose job satisfaction scores follow a Normal
distribution with standard deviation s = 60. Data from
18 workers gave a sample mean score of 17. If the null
hypothesis is µ0 = 0, the test statistic is:
x  μ0
17  0
z

 1.20
σ
60
n
18
BPS - 3rd Ed.
Chapter 14
22
Case Study II
Studying Job Satisfaction
For test statistic z = 1.20 and alternative hypothesis
Ha: m ≠ 0, the P-value would be:
P-value = P(Z < -1.20 or Z > 1.20)
= 2 P(Z < -1.20) = 2 P(Z > 1.20)
= (2)(0.1151) = 0.2302
If H0 is true, there is a 0.2302 (23.02%) chance that we
would see results at least as extreme as those in the
sample. Therefore do not have good evidence against
H0 and in favor of Ha.
BPS - 3rd Ed.
Chapter 14
23
Case Study II
Studying Job Satisfaction
BPS - 3rd Ed.
Chapter 14
24
Case Study II
Studying Job Satisfaction
1.
Hypotheses:
2.
Test Statistic:
H 0: m = 0
H a: m ≠ 0
z
x  μ0
σ

17  0
60
n
3.
4.
 1.20
18
P-value: P-value = 2P(Z > 1.20) = (2)(1 – 0.8849) = 0.2302
Conclusion:
Since the P-value is larger than a = 0.10, there is not sufficient
evidence that mean job satisfaction of assembly workers differs
when their work is machine-paced rather than self-paced.
BPS - 3rd Ed.
Chapter 14
25
Confidence Intervals & Two-Sided Tests
A level a two-sided significance test
rejects the null hypothesis H0: m = m0
exactly when the value m0 falls outside a
level 1 – a confidence interval for m.
BPS - 3rd Ed.
Chapter 14
26
Case Study II
Studying Job Satisfaction
A 90% confidence interval for m is:
xz
 σ
n
 17  1.645
60
 17  23.26
18
 6.26 to 40.26
Since m0 = 0 is in this confidence interval, it is plausible that
the true value of m is 0. Thus, there is insufficient evidence
(at a = 0.10) that the mean job satisfaction of assembly
workers differs when their work is machine-paced rather
than self-paced.
BPS - 3rd Ed.
Chapter 14
27