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Putting Statistics to Work
Copyright © 2011 Pearson Education, Inc.
Unit 6D
Statistical Inference
Copyright © 2011 Pearson Education, Inc.
Slide 6-3
6-D
Statistical Significance
A set of measurements or observations in a statistical
study is said to be statistically significant if it is
unlikely to have occurred by chance.
Common levels of significance:


At the 0.05 level – The probability of an observed
difference occurring by chance is 1 in 20 or less.
At the 0.01 level – The probability of an observed
difference occurring by chance is 1 in 100 or less.
Copyright © 2011 Pearson Education, Inc.
Slide 6-4
6-D
Margin of Error and Confidence Intervals
Suppose you draw a single sample of size n from a
large population and measure its sample proportion.

The margin of error for 95% confidence is
1
margin of error 
n

The 95% confidence interval is found by
subtracting and adding the margin of error from
the sample proportion.
 You can be 95% confident that the true
population proportion lies within this interval.
Copyright © 2011 Pearson Education, Inc.
Slide 6-5
6-D
Margin of Error and Confidence Intervals
Example: A survey of 1200 people finds that 47%
plan to vote for Smith for governor.
Find the margin of error.
1
1

 0.029
n
1200
Find the 95% confidence interval for the survey.
47% – 2.9% = 44.1%
47% + 2.9% = 49.9%
We can be 95% confident that the true proportion of
people who plan to vote for Smith is between
44.1% and 49.9%.
Copyright © 2011 Pearson Education, Inc.
Slide 6-6
6-D
Hypothesis Testing

The null hypothesis claims a specific value for a
population parameter. It takes the form
null hypothesis: population parameter = claimed value

The alternative hypothesis is the claim that is
accepted if the null hypothesis is rejected.
Copyright © 2011 Pearson Education, Inc.
Slide 6-7
6-D
Outcomes of a Hypothesis Test
A hypothesis is a statement regarding a characteristic of one or
more populations.

Rejecting the null hypothesis
→ We have evidence that supports the
alternative hypothesis.

Not rejecting the null hypothesis
→ We lack sufficient evidence to support the
alternative hypothesis.
Copyright © 2011 Pearson Education, Inc.
Slide 6-8
Examples of Claims Regarding a
Characteristic of a Single Population

6-D
In 2008, 62% of American adults regularly volunteered their
time for charity work. A researcher believes that this percentage
is different today.
Source: ReadersDigest.com poll created on 2008/05/02
10-9
Slide 6-9
6-D
Examples of Claims Regarding a
Characteristic of a Single Population

In 2008, 62% of American adults regularly volunteered their time
for charity work. A researcher believes that this percentage is
different today.

According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.
10-10
Slide 6-10
6-D
Examples of Claims Regarding a
Characteristic of a Single Population

In 2008, 62% of American adults regularly volunteered their time
for charity work. A researcher believes that this percentage is
different today.

According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.

Using an old manufacturing process, the standard deviation of the
amount of wine put in a bottle was 0.23 ounces. With new
equipment, the quality control manager believes the standard
deviation has decreased.
10-11
Slide 6-11
6-D
CAUTION!
We test these types of statements using sample data
because it is usually impossible or impractical to gain
access to the entire population. If population data are
available, there is no need for inferential statistics.
10-12
Slide 6-12
6-D
Alternative Hypothesis
The alternative hypothesis, denoted H1, is
a statement that we are trying to find
evidence to support. In this chapter, it
will be a statement regarding the value of
a population parameter.
10-13
Slide 6-13
6-D
Forming Hypotheses
For each of the following claims, determine the null and alternative
hypotheses.
a)
In 2008, 62% of American adults regularly volunteered their
time for charity work. A researcher believes that this
percentage is different today.
b)
According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.
c)
Using an old manufacturing process, the standard deviation of
the amount of wine put in a bottle was 0.23 ounces. With new
equipment, the quality control manager believes the standard
deviation has decreased.
10-14
Slide 6-14
Solution a)
a)
6-D
In 2008, 62% of American adults regularly volunteered
their time for charity work. A researcher believes that this
percentage is different today.
The hypothesis deals with a population proportion, p. If
the percentage participating in charity work is no different
than in 2008, it will be 0.62 so the null hypothesis is H0:
p=0.62.
Since the researcher believes that the percentage is
different today, the alternative hypothesis is: H1: p≠0.62.
Slide 6-15
6-D
Solution b)
b)
According to a study published in March, 2006 the mean
length of a phone call on a cellular telephone was 3.25
minutes. A researcher believes that the mean length of a
call has increased since then.
The hypothesis deals with a population mean. If the mean
call length on a cellular phone is no different than in 2006,
it will be 3.25 minutes so the null hypothesis is H0 =3.25.
Since the researcher believes that the mean call length has
increased, the alternative hypothesis is: H1 > 3.25.
Slide 6-16
Solution c)
c)
6-D
Using an old manufacturing process, the standard
deviation of the amount of wine put in a bottle was 0.23
ounces. With new equipment, the quality control manager
believes the standard deviation has decreased.
The hypothesis deals with a population standard deviation.
If the standard deviation with the new equipment has not
changed, it will be 0.23 ounces so the null hypothesis is H0
= 0.23.
Since the quality control manager believes that the
standard deviation has decreased, the alternative
hypothesis is: H1 < 0.23.
Slide 6-17
6-D
CAUTION!
We never “accept” the null hypothesis because
without having access to the entire population, we
don’t know the exact value of the parameter stated in
the null hypothesis. Rather, we say that we do not
reject the null hypothesis. This is just like the court
system. We never declare a defendant “innocent”,
but rather say the defendant is “not guilty”.
10-18
Slide 6-18
6-D
Stating Conclusions
According to a study published in March, 2006 the mean
length of a phone call on a cellular telephone was 3.25
minutes. A researcher believes that the mean length of
a call has increased since then.
a)
Suppose the sample evidence indicates that the null
hypothesis should be rejected. State the wording of
the conclusion.
b)
Suppose the sample evidence indicates that the null
hypothesis should not be rejected. State the wording
of the conclusion.
6-D
Solution a)
a)
Suppose the sample evidence indicates that the null
hypothesis should be rejected. State the wording of
the conclusion.
The statement in the alternative hypothesis is that the
mean call length is greater than 3.25 minutes. Since the
null hypothesis (H0 = 3.25) is rejected, we conclude that
there is sufficient evidence to conclude that the mean
length of a phone call on a cell phone is greater than 3.25
minutes.
10-20
Slide 6-20
6-D
Solution b)
b) Suppose the sample evidence indicates that the null
hypothesis should not be rejected. State the wording
of the conclusion.
Since the null hypothesis (H0 = 3.25) is not rejected, we
conclude that there is insufficient evidence to conclude
that the mean length of a phone call on a cell phone is
greater than 3.25 minutes. In other words, the sample
evidence is consistent with the mean call length equaling
3.25 minutes.
10-21
Slide 6-21
6-D
Hypothesis Test Decisions
Compare the actual sample result to the result
expected if the null hypothesis is true.
If the chance of a sample result at least as extreme
as the observed result is

less than 1 in 100
→ strong evidence to reject the null hypothesis

less than 1 in 20
→ moderate evidence to reject the null hypothesis

greater than 1 in 20
→ not sufficient evidence to reject the null hypothesis
Copyright © 2011 Pearson Education, Inc.
Slide 6-22
6-D
Hypothesis Test Example
Suppose the sample of 36 calls resulted in a sample mean
of 3.56 minutes with a standard deviation of 0.13. Do
the results of this sample suggest that the researcher is
correct? In other words, would it be unusual to obtain a
sample mean of 3.56 minutes from a population whose
mean is 3.25 minutes? What is convincing or
statistically significant evidence?
Slide 6-23
6-D
Statistical Significance
When observed results are unlikely under
the assumption that the null hypothesis is
true, we say the result is statistically
significant. When results are found to be
statistically significant, we reject the null
hypothesis.
10-24
Slide 6-24
6-D
Hypothesis Test Example (cont.)
Recall that our simple random sample of 36 calls resulted
in a sample mean of 3.56 minutes with standard
deviation of 0.13. Thus, the sample mean is
3.56 - 3.25
z=
= 2.38
0.13
standard deviations above the hypothesized mean of
3.25 minutes.
Therefore, using our criterion, we would reject the null
hypothesis and conclude that the mean cellular call
length is greater than 3.25 minutes.
Slide 6-25
6-D
Standard Scores and Percentiles
Copyright © 2011 Pearson Education, Inc.
Slide 6-26
6-D
Hypothesis Test Example (cont.)
Why does it make sense to reject the null hypothesis if the
sample mean is more than 2 standard deviations above the
hypothesized mean?
Slide 6-27
6-D
Hypothesis Test Example (cont.)
If the null hypothesis were true, then
97.72% of all sample means will be less
than 2 standard deviations above 3.25 or
3.25 + 2(0.13) = 3.51.
Slide 6-28
6-D
Hypothesis Test Example (cont.)
Because sample means greater than 3.51 are unusual if
the population mean is 3.25, we are inclined to believe
the population mean is greater than 3.25.
10-29
Slide 6-29
6-D
Assignment
p. 410 – 412 9-20, 22, 27, 30, 33, 36, 39, 40, 45 , 46
Copyright © 2011 Pearson Education, Inc.
Slide 6-30