7.0: Hypothesis Testing
Note: For access to statistical tables and equations, please click "Help Files".
As we have seen throughout this course, we often desire to make decisions about
population characteristics based on sample data. In this module, we will make decisions
by first making an assumption or hypothesis about a certain characteristic of the
population, and then test to see if the hypothesis is true. This is called hypothesis
testing.
As we progress through this module, please be patient. We will be building as we
continue through the module. The first part of the module will lay the foundation for
hypothesis testing. Then, we will learn some of the tools required for hypothesis testing.
Finally, near the end of the module, we will do complete hypothesis testing problems.
Skip To Content
7.1: Basic Definitions
First, we will give some definitions:
Null hypothesis, H0, is the assumption or hypothesis that is to be tested. We assume that
the null hypothesis is true until proven otherwise.
Alternate hypothesis, H1, is the hypothesis for which we are trying to gather evidence.
If we prove that the alternate hypothesis is true, then we reject the null hypothesis. If we
cannot prove that the alternate hypothesis is true, then we do not reject the null
hypothesis.
In order to understand how we might apply these definitions, let’s consider a criminal
trial. In any criminal trial, the defendant is assumed to be innocent, but the prosecutor
will try to prove that the defendant is guilty. So, we may assign the null hypothesis and
alternate hypothesis to a criminal trial in the following way:
Null hypothesis, H0: The person is innocent.
Alternate hypothesis, H1: The person is guilty.
During the trial, the prosecution presents evidence (or, in mathematical terms, we might
think of this as sample data). The jury, or the judge, weighs the evidence (the sample
data) and must decide the case. The two possible conclusions are 1) to reject the null
hypothesis (find that the person is guilty), or, 2) to not reject the null hypothesis (find
that the person is not guilty).
,Example 7.1
Define null and alternate hypothesis in each of the following:
A) The cereal company claims that it puts 16 ounces of cereal in each box of corn flakes.
In particular, the company claims that mean weight of all of its cereal boxes is 16 ounces
with a standard deviation of .212 ounces. You believe that the company is not putting
quite 16 ounces in the boxes.
B) The average SAT score for math this year was 514. The principal in a certain high
school believes that his school’s math SAT scores are higher.
C) In 2005, your company did a study in order to determine how long it took for a
package to be delivered. It found that it took an average of 4 days for a 10-pound
package to get from Pittsburgh to Los Angeles. The new shipping supervisor believes
that that the average delivery time is different today.
Solution
A) Since the cereal company claims to have 16 ounces of cereal in each box, we set the
null hypothesis at:
H0: μ = 16 ounces.
You believe that the boxes may be short of 16 ounces. So, we set the alternative
hypothesis as:
H1: μ < 16 ounces.
B) The average math SAT score was 514, so we set that as the null hypothesis:
H0: μ = 514
The principal believes that the math SAT score in his school is higher than 514, so we set
the alternative hypothesis at:
H1: μ > 514 .
C) Based on the information that we have from 2005, it takes 4 days for a 10 pound
package to get from Pittsburgh to Los Angeles, so we set that as the null hypothesis:
H0: μ = 4 days.
The new shipping supervisor believes that that the average delivery time is different
today. It is important to note that the shipping supervisor believes that the average
shipping time is different today, not necessarily higher or lower, but different. So, we set
the null hypothesis as:
, H1: μ 4 days.
Let’s examine the alternate hypotheses in A), B), and C) above. Notice that for the
alternative hypothesis in A), the mean is less than the mean given in the null hypothesis.
In B), the alternative hypothesis has the mean greater than the mean given in the null
hypothesis. The alternative hypothesis in C) has the mean different from the mean given
in the null hypothesis. So, we will reject the null hypothesis in C), if it can be proven that
the mean is less than or greater than the mean given in the null hypothesis.
As illustrated in this last example, there are three way to set up the null and alternative
hypothesis. These three ways may be formally stated as follows:
For any parameter, p and any number c, (in many of the problems that we do in this
module, p will be the mean µ).
1) Left-tailed test
H0: p = c
H1: p < c
2) Right-tailed test
H0: p = c
H1: p > C
3) Two-tailed test
H0: p = c
H1: p ≠ c
Practice Problems:
1. In a certain suburb, the mean house price in 2009 was $ 225,000. A real estate agent
believes that the mean house price in that suburb has increased since 2009. Please state
the null and alternative hypothesis.
2. According to babycenter.com, the mean height of a six year old boy is 42 inches. A
school counselor believes that the mean height of the six year old boys in her school is
less than 42 inches. Please state the null and alternative hypothesis.
3. In 2012, it was reported that McDonald’s had 62 million customers per day. A
researcher believes that the number of McDonald’s customers per day is different now.
Please state the null and alternative hypothesis.
Note: For access to statistical tables and equations, please click "Help Files".
As we have seen throughout this course, we often desire to make decisions about
population characteristics based on sample data. In this module, we will make decisions
by first making an assumption or hypothesis about a certain characteristic of the
population, and then test to see if the hypothesis is true. This is called hypothesis
testing.
As we progress through this module, please be patient. We will be building as we
continue through the module. The first part of the module will lay the foundation for
hypothesis testing. Then, we will learn some of the tools required for hypothesis testing.
Finally, near the end of the module, we will do complete hypothesis testing problems.
Skip To Content
7.1: Basic Definitions
First, we will give some definitions:
Null hypothesis, H0, is the assumption or hypothesis that is to be tested. We assume that
the null hypothesis is true until proven otherwise.
Alternate hypothesis, H1, is the hypothesis for which we are trying to gather evidence.
If we prove that the alternate hypothesis is true, then we reject the null hypothesis. If we
cannot prove that the alternate hypothesis is true, then we do not reject the null
hypothesis.
In order to understand how we might apply these definitions, let’s consider a criminal
trial. In any criminal trial, the defendant is assumed to be innocent, but the prosecutor
will try to prove that the defendant is guilty. So, we may assign the null hypothesis and
alternate hypothesis to a criminal trial in the following way:
Null hypothesis, H0: The person is innocent.
Alternate hypothesis, H1: The person is guilty.
During the trial, the prosecution presents evidence (or, in mathematical terms, we might
think of this as sample data). The jury, or the judge, weighs the evidence (the sample
data) and must decide the case. The two possible conclusions are 1) to reject the null
hypothesis (find that the person is guilty), or, 2) to not reject the null hypothesis (find
that the person is not guilty).
,Example 7.1
Define null and alternate hypothesis in each of the following:
A) The cereal company claims that it puts 16 ounces of cereal in each box of corn flakes.
In particular, the company claims that mean weight of all of its cereal boxes is 16 ounces
with a standard deviation of .212 ounces. You believe that the company is not putting
quite 16 ounces in the boxes.
B) The average SAT score for math this year was 514. The principal in a certain high
school believes that his school’s math SAT scores are higher.
C) In 2005, your company did a study in order to determine how long it took for a
package to be delivered. It found that it took an average of 4 days for a 10-pound
package to get from Pittsburgh to Los Angeles. The new shipping supervisor believes
that that the average delivery time is different today.
Solution
A) Since the cereal company claims to have 16 ounces of cereal in each box, we set the
null hypothesis at:
H0: μ = 16 ounces.
You believe that the boxes may be short of 16 ounces. So, we set the alternative
hypothesis as:
H1: μ < 16 ounces.
B) The average math SAT score was 514, so we set that as the null hypothesis:
H0: μ = 514
The principal believes that the math SAT score in his school is higher than 514, so we set
the alternative hypothesis at:
H1: μ > 514 .
C) Based on the information that we have from 2005, it takes 4 days for a 10 pound
package to get from Pittsburgh to Los Angeles, so we set that as the null hypothesis:
H0: μ = 4 days.
The new shipping supervisor believes that that the average delivery time is different
today. It is important to note that the shipping supervisor believes that the average
shipping time is different today, not necessarily higher or lower, but different. So, we set
the null hypothesis as:
, H1: μ 4 days.
Let’s examine the alternate hypotheses in A), B), and C) above. Notice that for the
alternative hypothesis in A), the mean is less than the mean given in the null hypothesis.
In B), the alternative hypothesis has the mean greater than the mean given in the null
hypothesis. The alternative hypothesis in C) has the mean different from the mean given
in the null hypothesis. So, we will reject the null hypothesis in C), if it can be proven that
the mean is less than or greater than the mean given in the null hypothesis.
As illustrated in this last example, there are three way to set up the null and alternative
hypothesis. These three ways may be formally stated as follows:
For any parameter, p and any number c, (in many of the problems that we do in this
module, p will be the mean µ).
1) Left-tailed test
H0: p = c
H1: p < c
2) Right-tailed test
H0: p = c
H1: p > C
3) Two-tailed test
H0: p = c
H1: p ≠ c
Practice Problems:
1. In a certain suburb, the mean house price in 2009 was $ 225,000. A real estate agent
believes that the mean house price in that suburb has increased since 2009. Please state
the null and alternative hypothesis.
2. According to babycenter.com, the mean height of a six year old boy is 42 inches. A
school counselor believes that the mean height of the six year old boys in her school is
less than 42 inches. Please state the null and alternative hypothesis.
3. In 2012, it was reported that McDonald’s had 62 million customers per day. A
researcher believes that the number of McDonald’s customers per day is different now.
Please state the null and alternative hypothesis.
