Fundamentals of Hypothesis
Testing
Chapter 10
Created by Laura Ralston
Outline
• 10.1
• 10.2
• 10.3
• 10.4
•
•
•
•
10.5
10.6
10.7
10.8
Fundamentals of Hypothesis Testing
Hypothesis Testing for Means (Small
Samples)
Hypothesis Testing for Means (Large
Samples)
Hypothesis Testing for Population
Proportions ?
Types of Errors
Hypothesis Testing for Population Variance
Chi-Square Test for Goodness of Fit
Chi-Square Test for Association
Introduction
• Anyone can make a claim about a population
parameter
– “Four out of five dentists prefer StarBrite toothpaste”
– “Fewer than 1% of dogs attack people unprovoked”
– “In any given one-year period, ….about 18.8 million
American adults suffers from a depressive illness” –
Reader’s Digest
– “For cars, this Corporate Average Fuel Economy has
been 27.5 miles per gallon since 1990….” USA Today
• People make decisions everyday based on
such claims.
– Some decisions are insignificant (which toothpaste
to buy)
– Others are very important!
• Whether to use a new prescription drug that has just
come onto the market
• Purchase a car based on reported safety ratings
• Are public safety campaigns working?
• Which candidate is likely to win an election?
Hypothesis Testing
• One statistical process that sets a uniform
standard for evaluating claims about a population
parameter
• Foundation for hypothesis testing is the RARE
EVENT RULE: if, under a given assumption, the
probability of a particular observed event is
exceptionally small, we conclude that the
assumption is probably not correct (Gender
selection example from Triola)
Step 1: State the hypotheses in mathematical
terms
Null Hypothesis, H0
• Claim about a population
parameter
• Always contains “equals”
Alternative Hypothesis, H1 or Ha
• Claim about a population
parameter
• Opposite of the null
hypothesis
Indicate which hypothesis is stated in the
given problem (o.c.). This information is
needed later.
Common Phrases
>
Is greater than
Is above
Is higher than
Is longer than
Is bigger than
Is increased
<
Is less than
Is below
Is lower than
Is shorter than
Is smaller than
Is decreased or reduced from
>
Is greater than or equal to
Is at least
Is not less than
<
Is less than or equal to
Is at most
Is not more than
=
Is equal to
Is exactly the same as
Has not changed from
Is the same as
≠
Is not equal to
Is different from
Has changed from
Is not the same as
Step 2: Determine the critical value(s)
• Based on the level of significance, a, which will be
given in the problem and using normal distribution
• Critical value(s) separates “guilty” and “not guilty”
• Use calculator: invNorm command
• “Type” of test is determined by Ha which will
determine the critical region(s) and ultimately the
critical value(s)
– Right-Tailed testif > or >
(picture)
– Left-Tailed test
if < or <
(picture)
– Two-tailed test
if ≠
(picture, a/2)
Step 3: Calculate test statistic
• Test statistic will vary depending on the
parameter and sample size (z, t, or chi-square)
• Used to make a decision about the null
hypothesis, “evidence”
• Calculated using sample data (given in
problem)
𝑧=
𝑥−𝜇
𝑠
𝑛
Step 4: Make a decision
• Variety of methods (traditional, p-value,
confidence intervals)
• Two possibilities
– Reject the Null Hypothesis -assumption is not
valid based on this set of sample data
– Fail to reject the Null Hypothesis (never “accept”)assumption is valid based on this set of sample
data
Step 5: State final conclusion in nontechnical terms
• Conclusion should be well-worded, reference
the original claim, and free of mathematical
symbols.
• See flow chart on next slide to help determine
wording
Yes
YES (original claim
contains equality)
Do you reject H0 ?
No
Does the original claim
contain the condition of
equality?
NO (original claim does
not contain equality)
“There is sufficient evidence
to warrant rejection of the
claim that …..(original claim)
Yes
“There is not sufficient
evidence to warrant
rejection of the claim that
…(original claim).”
“The sample data support
the claim that ….(original
claim).”
Do you reject H0?
No
“There is not sufficient
sample evidence to support
the claim that ….(original
claim)”
Example 1
• In a sample of 27 blue M&Ms with a mean
weight of 0.8560 g. Assume that s is known to
be 0.0565 g. Consider a hypothesis test that
uses a 0.05 significance level to test the claim
that the mean weight of all M&Ms is equal to
0.8535 g (the weight necessary so that bags of
M&Ms have the weight printed on the
packages).
Example 2
• In a sample of 106 body temperatures with a
mean of 98.20°F. Assume that s is known to
be 0.62°F. Consider a hypothesis test that
uses a 0.05 significance level to test the claim
that the mean body temperature of the
population is less than 98.6°F.
Example 3
• The average production of peanuts in the
state of Virginia is 3000 pounds per acre. A
new plant food has been developed and is
tested on 60 individual plots of land. The
mean yield with the new plant food is 3120
pounds of peanuts per acre with a standard
deviation of 578 pounds. At a 0.05
significance level, can one conclude that the
average production has increased?
Example 4
• A researcher claims that the yearly
consumption of soft drinks per person is 52
gallons. In a sample of 50 randomly selected
people, the mean of the yearly consumption
was 56.3 gallons. The standard deviation of
the sample was 3.5 gallons. At a 0.01
significance level, is the researcher’s claim
valid?
Assignment
• Use traditional method ---modeled in above
examples
• Page 498 #25-30