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MTL Module 6B Unit 4
Math 490 Module 6B
Unit 4: Hypothesis Testing with One Sample
Goals: understand hypothesis testing; introduce Minitab® software
Read the Chapter or unit on the Hypothesis Testing with One Sample in your primary AP
Statistics textbook, and study sections 7.5, 8.4 & 8.5 in the AMSCO text.
Historical Note: The t distribution was formulated in 1908 by an Irish brewing employee named
William Sealey Gosset. Gosset was involved in researching new methods of manufacturing ale.
Gosset published his finding using the pseudonym Student; hence, the t distribution is sometimes
called the Student’s distribution.
The Basic Rule for Hypothesis Testing: “If, under a given assumption, the probability of a
particular observed event is exceptionally small, we conclude that the assumption is probably not
correct.” (Elementary Statistics 9e, Triola) Using this rule, we test a hypothesis by analyzing a
sample data in an attempt to distinguish between results that can easily occur by chance and
results that are highly unlikely to occur by chance. When we get highly unlikely results, we
conclude that the hypothesis is not true.
Instead of beginning with a sequence of mechanical steps, begin hypothesis testing with an
overview of the basic concept used. Focus on the issue of significance: Do the sample results
differ from the hypothesis by an amount that is statistically significant?” The following
assignment will help you focus on statistical significance.
Assignment:
4.1. Develop or find an example, which you can use in your classroom, which illustrates
the fundamental idea/basic approach of hypothesis tests. Do not write a hypothesis
or any of the typical steps in hypothesis testing. Instead, write a simple narrative of
a claim, and give two different sample results: one that might easily occur by
chance, and another result that would be highly unlikely to occur by chance.
Consider how you will explain statistical significance to your class as you develop
the example.
Components of a formal Hypothesis Test
The null hypothesis, H0, is a statement that the value of the population parameter is equal to a
claimed value. Some texts use the symbols ≤ or ≥ in the null hypothesis, but most professional
journals use only the =. We test the null hypothesis by assuming it is true, and reach a
conclusion to reject it or fail to reject it.
The alternate hypothesis, H1 or Ha, is the statement that the parameter has a value that
somehow differs from the null hypothesis. If you are conducting a study and want to use a
hypothesis test to support your claim, the claim should be worded so that it becomes the alternate
hypothesis, because you cannot use a hypothesis test to support a claim that a parameter is equal
to a specified value. However, someone else’s claim may become either the null or the alternate
hypothesis. The best way to form the null and alternate hypothesis is to write the claim
symbolically, and then write the opposite of the claim; the statement containing the equality is
the null hypothesis.
MTL Module 6B Unit 4
Assignment:
4.2. Write or find a set of claims that you can give students to practice expressing the
corresponding null and alternate hypotheses in symbolic form.
The test statistic is a value computed from the sample data, and used in making the decision
about the rejection of the null hypothesis. The test statistic is found by converting a sample mean
to a z or t score, sample proportion p to a z score, or standard deviation s to a X2 score.
The P-value is the probability associated with the test statistic.
The critical region (rejection region) is the set of values of the test statistics that cause us to
reject the null hypothesis. The significance level is the probability that the test statistic will fall
in the critical region when the hypothesis is true. If the test statistic falls in the critical region, we
will reject the null hypothesis, and the probability of rejecting it when it may be true is a type I
error called .
The critical value is any value that separates the critical region from the values that do not lead
to rejection of the null hypothesis.
Read about the test statistic and critical values in your primary text. Read about two-tailed,
left-tailed, and right-tailed tests. Read about P-values.
Assignment:
4.3. Write at least ten practice problems to use with your class asking students look up
critical values. Use a significance level of 0.05, write a set of alternate hypotheses
using sample mean, proportion, and standard deviation, and <, >, and ≠. Ask
students to find the P-value for each problem.
Decision: The standard procedure of hypothesis testing is that we always test the null hypothesis
and conclude one of the following:
1. Reject the null hypothesis if the test statistic falls within the critical region.
2. Fail to reject the null hypothesis it the test statistic does not fall within the critical region.
Published research often uses the P-value method:
1. Reject the null hypothesis if the P-value < .
2. Fail to reject the null hypothesis if the P-value ≥ .
3. State the P-value and leave the decision for the reader.
Conclusion: Students typically have difficulty writing a correct statement of the final
conclusion. The conclusion should address the original claim, and precise wording is important.
Students often don’t understand the difference between “accept or support” and “fail to reject.”
Assignment:
4.4. Explain the difference between “accept or support” and “fail to reject.” You may
find an example in your text, or create one, but your response should be in your own
words.
MTL Module 6B Unit 4
Read about Type I and Type II errors, and the power of a hypothesis test in your primary text.
Read section 7.6 (Type I and Type II errors) in the Amsco supplementary text.
The AP test will most likely not ask students to calculate statistical power, but includes the
concept of Type I and Type II errors and power in their content guide. The following multiple
choice question is taken from the 2002-2003 AP guide.
The corn rootworm is a pest that can cause significant damage to corn, resulting in a reduction in
yield and thus in farm income. A farmer will examine a random sample of plants from a field in
order to decide whether or not the number of corn rootworms in the whole field is at a dangerous
level. If the farmer concludes that it is, the field will be treated. The farmer is testing the null
hypothesis that the number of corn rootworms is not at a dangerous level against the alternative
hypothesis that the number is at a dangerous level. Suppose that the number of corn rootworms
in the whole field actually is at a dangerous level.
Which of the following is equal to the power of the test?
a) The probability that the farmer will decide to treat the field
b) The probability that the farmer will decide not the treat the field
c) The probability that the farmer will fail to reject the null hypothesis
d) The probability that the farmer will reject the alternative hypothesis
e) The probability that the farmer will not get a statistically significant result
StatDisk, the TI-83+, Excel, and various interactive websites will give results for a hypothesis
test of sample means, proportions, and standard deviations, and you are encouraged to try all of
these applications using problems in your primary text. Most texts will have supplementary
information and examples of these applications.
Minitab® was originally published for educational use, but in current years is marketed more for
industrial and commercial applications, although the website still supports educational use.
Minitab no longer produces a version for Mac OS, but Release 14 will run under Windows
simulation. Minitab instructions and printouts can be found in many current AP Statistics
textbooks, and you may find it helpful to read these instructions before using Minitab. Because
Minitab is found in so many AP Statistics texts, it is important for AP teachers to experience
using it. A fully functional 30-day trial version of Minitab Release 14 can be downloaded at:
http://www.minitab.com/products/minitab/14/demo/
Assignment:
4.5.
To get acquainted with Minitab, use this data for a hypothesis test where the
alternative hypothesis is that the standard deviation is greater than 8.
25, 42, 12, 30, 16, 38, 5, 9, 8, 15, 10, 14, 18, 12, 27
Self-test using the review exercises at the end of chapter 8 in the AMSCO text, and practice
equivalent problems in your primary AP Statistics text.
MTL Module 6B Unit 4
The following problem is one of my favorites (I do not know in what textbook the problem
originally appeared), and I present it here for your practice. As you work through the problem,
consider the assumptions you must make to answer the questions.
“Television networks typically project a winner in a political contest long before the votes have
been completely tabulated. In the current election, the Democrat is favored to win over the
Republican opponent. You are watching the broadcast of the evening election returns, but you
would like to go to sleep as soon as you are confident of the outcome of the race.
(a)
State the assertion that the Democrat will not win as a null hypothesis. Then state the
alternative hypothesis. (Note: It takes more than 50 percent of the vote to win.)
(b)
At 9 p.m., a newscaster announces that 100 ballots have been counted and that the
Democrat has received 55 percent of the votes. If you conclude that the Democrat has
won - that is, if you reject the null hypothesis at this point and go to sleep - what risk do
you face of waking up the next morning to find that the Republican has been elected?
(Assume that the 100 ballots counted are a simple random sample of the ballots cast.)
(c)
At 10 p.m., the newscaster announces that 2,500 ballots have been counted an that 51.6
percent of the votes favor the Democrat. How probable is it that these results would arise
if the null hypothesis were true?
(d)
At 11 p.m., you hear the newscaster say that 10,000 votes have been tabulated, but you
do not hear what proportion of these votes has been cast for the Democrat. What
proportion would lead you to reject the null hypothesis and go to sleep at this point, if
you are willing to accept a 1% risk of waking up to find that the Republican has been
elected?”
Assignment:
4.6.
Select a problem from a problem set in the chapter on one-sample hypothesis test in
your primary text. Copy the problem, and then show how to do the hypothesis test
using the TI-83+, Statdisk or Minitab, Excel, and/or another software application of
your choice. Three different applications are required. Briefly compare and contrast
the applications for this test.
4.7.
Do an internet search for a sample data set, give the URL, and if no claim is made
on the site, make a claim about the data. Then write a hypothesis test for the claim
using both Minitab and one other application. Write a good and a badly-written
conclusion.
4.8.
Find an internet site for interactive testing of a parameter. Give the URL, and give
evidence that you have tried hypothesis testing using the features of the site.
4.9.
Write an activity you might use to introduce and explain hypothesis testing to a high
school AP class. You may write your response as a lesson plan, or submit a
worksheet(s) you have developed. You must use at least two different technologies
with your plan/worksheet.
END OF UNIT 4