INTRODUCTORY STATISTICS
Chapter 9 HYPOTHESIS TESTING WITH ONE SAMPLE:
SINGLE MEAN AND SINGLE PROPORTION
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SEC. 9.2: NULL AND ALTERNATIVE HYPOTHESIS
You can use a hypothesis
test to decide if a dog
breeder’s claim that every
Dalmatian has 35 spots is
statistically sound. (Credit:
Robert Neff)
Hypothesis testing consists of two contradictory hypotheses or statements, a
decision based on the data, and a conclusion. To perform a hypothesis test, a
statistician will:
• Set up two contradictory hypotheses.
• Collect sample data (in homework problems, the data or summary statistics
will be given to you).
• Determine the correct distribution to perform the hypothesis test.
• Analyze sample data by performing the calculations that ultimately will allow
you to reject or decline to reject the null hypothesis.
• Make a decision and write a meaningful conclusion.
NULL AND ALTERNATIVE HYPOTHESIS
The actual test begins by considering two hypotheses. They are
called the null hypothesis and the alternative hypothesis. These
hypotheses contain opposing viewpoints.
H0: The null hypothesis: It is a statement about the population that
either is believed to be true or is used to put forth an argument unless
it can be shown to be incorrect beyond a reasonable doubt.
Ha: The alternative hypothesis: It is a claim about the population
that is contradictory to H0 and what we conclude when we reject H0.
NULL AND ALTERNATIVE HYPOTHESIS
Note: Null hypothesis ALWAYS has an equal in it and alternative hypothesis does not.
When talking about the mean, state each in terms of 𝜇.
When talking about a proportion, state each in terms of p.
PROCESS
1. Find the population information.
2. Determine if you have a mean or a proportion
3. Determine the operators (math symbols). Look for key terms like
“more than,” “fewer than,” “at least,” etc.
4. Make sure that the null and alternative hypotheses contradict
each other.
STATING NULL AND ALTERNATIVE HYPOTHESES
The General Manager of Fresh'n'Easy air conditioners tells an investigative reporter
that at least 85% of its customers are "completely satisfied" with their overall purchase
performance. State the null and alternative hypotheses.
𝐻0 : 𝑝 ≥ 0.85
(The proportion is greater than or equal to 85%)
𝐻𝐴 : 𝑝 < 0.85
(The proportion is less than 85%)
STATING NULL AND ALTERNATIVE HYPOTHESES
A student counsellor claims that first year Science students spend an average 3 hours
per week doing exercises in each subject. What hypotheses will be used by a lecturer
to test the claim?
𝐻0 : μ = 3
(The mean is equal to 3)
𝐻𝐴 : μ ≠ 3
(The mean is not equal to 3)
DETERMINING INFORMATION
A recent survey of college campuses across Ontario claims that students spend an
average of 2.7 hours a day using their cell phones. A random sample of 35 Durham
College students showed an average use of 2.9 hours a day, with a standard deviation
of 0.4 hours. State the null and alternative hypotheses to determine if Durham College
students use their cell phones more than the typical Ontario college student?
𝐻0 : μ ≤ 2.7
(The mean is less than or equal to 2.7)
𝐻𝐴 : μ > 2.7
(The mean is more than 2.7)
PRACTICE
In an issue of U. S. News and World Report, an article on school standards stated that
about half of all students in France, Germany, and Israel take advanced placement
exams and a third pass. The same article stated that 6.6% of U. S. students take
advanced placement exams and 4.4% pass. State the hypotheses to test if the
percentage of U. S. students who take advanced placement exams is more than 6.6%
𝐻0 : p ≤ 6.6
(The proportion is less than or equal to 6.6)
𝐻𝐴 : 𝑝 > 6.6
(The proportion is more than 6.6)
SEC. 9.3: OUTCOMES AND TYPE I AND II ERRORS
When you perform a hypothesis test, there are four possible outcomes depending on
the actual truth (or falseness) of the null hypothesis H0 and the decision to reject or not.
The outcomes are summarized in the following table:
DECISIONS…
The four possible outcomes in the table are:
1. The decision is not to reject H0 when H0 is true (correct decision).
2. The decision is to reject H0 when H0 is true (incorrect decision known as a Type I
error).
3. The decision is not to reject H0 when, in fact, H0 is false (incorrect decision known
as a Type II error).
4. The decision is to reject H0 when H0 is false (correct decision whose probability is
called the Power of the Test).
EXAMPLE
A car company claims that its cars passed a safety test at least 96% of
the time.
a)
State the null hypothesis.
b)
𝐻0 : 𝑝 ≥ 0.96
State the Type I error in complete sentences.
c)
We conclude the percentage is less than 96% when it is actually
greater or equal.
State the Type II error in complete sentences.
d)
We conclude the percentage is at least 96% when it is actually
less than that.
What could be a result of a Type I error?
e)
Spending unnecessary money on improving safety when the car
is actually safe enough.
What could be a result of a Type II error?
Assuming the car is safe when it is actually not.
ANOTHER EXAMPLE
A poll conducted by a city determines that the mean commute time for its
residents is less than 45 minutes.
a)
State the null hypothesis.
b)
𝐻0 : 𝜇 ≥ 45
State the Type I error in complete sentences.
c)
We conclude the average commute is less than 45 minutes when
it is actually at least 45 minutes.
State the Type II error in complete sentences.
d)
We conclude the average commute is at least 45 minutes when it
is actually shorter.
What could be a result of a Type I error?
e)
Commuters do not allow sufficient time to get to work and are late.
What could be a result of a Type II error?
Commuters overestimate how long it will take them to get to work
and arrive early.
PRACTICE:
63% of American adults get the recommended amount of sleep per
night.
a) State the null hypothesis.
b) State the Type I error in complete sentences.
c) State the Type II error in complete sentences.
d) What could be a result of a Type I error?
e) What could be a result of a Type II error?
SEC. 9.6: FULL HYPOTHESIS TESTS
The hypothesis test itself has an established process. This can be
summarized as follows:
1. Determine H0 and Ha. Remember, they are contradictory.
The alternative hypothesis, Ha, tells you if the test is left, right, or
two-tailed. It is the key to conducting the appropriate test. Ha never
has a symbol that contains an equal sign.
LEFT, RIGHT OR TWO TAILED TESTS
Depending on 𝐻𝐴 and the probability of obtaining our sample value,
we will reject the null hypothesis if it falls in the shaded area.
EXAMPLE
If we are testing the claim that Dalmatians have 35 spots, state the
alternative hypothesis, sketch a graph and decide if it is left, right or
two tailed.
Do the same for if the claim is that Dalmatians have more than 35
spots.
STEP 2: DETERMINE THE RANDOM VARIABLE.
This will be either 𝑋 or P’ depending on whether you are considering a
mean or a proportion.
STEP 3: DETERMINE THE DISTRIBUTION FOR THE
TEST.
A Student's t-test should be used if the data come from a simple,
random sample and the population is approximately normally
distributed, or the sample size is large, with an unknown standard
deviation.
The normal test will work if the data come from a simple, random
sample and the population is approximately normally distributed, or
the sample size is large, with a known standard deviation.
STEP 4: CALCULATE THE P-VALUE
The p-value is the probability that an event will happen purely by
chance assuming the null hypothesis is true. The smaller the p-value,
the stronger the evidence is against the null hypothesis.
STEP 5: MAKE A DECISION
Compare the preconceived α (significance level) with the p-value,
make a decision (reject or do not reject H0), and write a clear
conclusion using English sentences. If 𝛼 is not stated, use a 5%
significance level.
α > p-value, reject the null hypothesis
α ≤ p-value, do not reject the null hypothesis
USING YOUR CALCULATOR
There are hypothesis test functions in your calculator under the STAT,
TESTS menu.
For means,
If you are using the normal distribution, use the Z-Test.
If you are using the Student-t distribution, use the T-Test.
For proportions, use the 1-PropZ-Test.
EXAMPLE
A principal at a certain school claims that the students in his school
are above average intelligence. A random sample of thirty students IQ
scores have a mean score of 112. Is there sufficient evidence to
support the principal’s claim? The mean population IQ is 100 with a
standard deviation of 15. Assume a significance level of 0.01.
EXAMPLE
Toastmasters International cites a report by Gallop Poll that 40% of
Americans fear public speaking. A student believes that less than 40%
of students at her school fear public speaking. She randomly surveys
361 schoolmates and finds that 135 report they fear public speaking.
Conduct a hypothesis test to determine if the percent at her school is
less than 40% at the 1% significance level.
EXAMPLE
According to the N.Y. Times Almanac the mean family size in the U.S.
is 3.18. A sample of a college math class resulted in the following
family sizes:
5, 4, 5, 4, 4, 3, 6, 4, 3, 3, 5, 5, 6, 3, 3, 2, 7, 4, 5, 2, 2, 2, 3, 2
At α = 0.05 level, is the class’ mean family size greater than the
national average? Does the Almanac result remain valid? Why?
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