Chapter 9
Fundamentals of Hypothesis Testing:
One-Sample Tests

9.1: Hypothesis Testing Methodology
• Confidence Intervals were our first Inference
• Hypothesis Tests are our second Inference
• “Methodology” implies a series of steps:
1. Develop hypotheses
2. Determine decision rule
3. Calculate test statistic
4. Compare results from 2 and 3: make a decision
5. Write conclusion

Step 1: Develop hypotheses
• You will need to develop 2 hypotheses:
1. Null hypothesis
2. Alternative hypothesis
– Hypotheses concern the population parameter in question (ie “µ” or “π” or other)

The Null Hypothesis
• A theory or idea about the population parameter .
• Always contains some sort of equality.
• Very often described as the hypothesis of “no difference” or
“status quo.”
• H
0
: µ = 368

The Alternative Hypothesis
• An idea about a population parameter that is the opposite of the idea in the null hypothesis
• NEVER contains any sort of equality!
• H
1
: µ

368 (sometimes use H a
:)

Hypotheses
• Null and alternative hypotheses are mutually exclusive and collectively exhaustive.
• Our sample either contains enough information to reject the NULL hypothesis
OR the sample does not contain enough information to reject the null hypothesis.

Step 2: Decision Rule or
“Rejection Region”
• Always says something like “we shall reject H
0 for some extreme value of the test statistic.”
• The “Rejection Region” is the range of the test statistic that is extreme enough—so extreme that the test statistic probably would not occur IF the null hypothesis is true.
• Figure 9.1 shows the rejection region for a hypothesis test of the mean.
• The “critical value” is looked up based on the error rate that you are comfortable with.

Step 3: Calculating the Test
Statistic
• The test statistic depends on the sampling distribution in use. This depends on the parameter.
• This will be determined the same way it was in chapter 8.

Step 4 & 5: Decision and
Conclusion
• The decision is always either (1) reject
H
0 or (2) fail to reject H
0
. This is determined by evaluating the decision rule in step 2.
• The conclusion always says “At α =
0.05, there is ( in )sufficient information to say H
1
”

Alpha
• α is the probability of committing a Type I error: erroneously rejecting a true Null
Hypothesis.
• α is called “The Level of Significance”
• α is determined before the sample results are examined.
• α determines the critical value and rejection region(s).
• α is set at an acceptably low level.

• Beta is the probability of committing a Type II error: erroneously failing to reject a false null hypothesis.
• Beta depends on several factors and it cannot be arbitrarily set. Beta can be indirectly influenced.

Compliments of Alpha and Beta
• (1-α) is called the confidence coefficient.
This is what we used in Chapter 8.
• (1-beta) is called the Power of the test.
Power is the chance of rejecting a null hypothesis that ought to be rejected, ie a false null. Bigger is better. Power cannot be set directly.

9.2: z Test of Hypothesis for the mean
• Use this test ONLY for the mean and only when σ is known.
• There are two approaches:
– critical value approach
– “p” value approach

Critical Value Approach
Remember your methodology (steps):
1. Create hypotheses
2. Create decision rule
– depends on α
– depends on distribution
3. Calculate test statistic
4. State the result
5. State the managerial conclusion

Hypotheses
• The discussion in 7.2 assumes a twotail test because the sample mean might be extremely large or extremely small.
– Either one would make you think the null hypothesis is wrong.

• The standard approach requires that the value of α be divided evenly between the tail areas.
• These tail areas are called the
“rejection region.”

Conclusion
• See Step 6 on pages 308-309.

“p-value” approach
• Rewrite the decision rule to say, “we will reject the null hypothesis if the ‘p-value’ is less than the value of α .”
• “p-value” definition, page 309.
• “p-value” is called the observed level of significance.
• Excel--most statistical software--does a good job of this (that’s why it’s a popular approach).

Estimation and Hypothesis Testing
• The two inferences are closely related.
• Estimation answers the question “what is it?”
• Hypothesis Testing answers the question “is it ______ than some number?”
• See page 312.

9.3: One-Tail Tests
• The rejection region is one single area.
• Sometimes called a directional test.
• Mechanics:
– see the text example
• Problem identification:
– hypothesis test or interval estimation?
– One-tail or Two-tail?
– If One-tail, which is the null?

• Page 314, the “milk problem.”
• Are we buying “watered-down milk” ?
– Watered-down milk freezes at a colder temperature than normal milk.
– What are the null and alternative hypotheses?
– Hint: what do you want to conclude?
– Hint: what is the hypothesis of action?
– Hint: what is the hypothesis of status quo?

Mechanics of the One-tailed test
• Different hypotheses.
• Different decision rule/rejection region.
• Different “p-value” or observed significance of observed level of significance.

Consider Problem 9.44, page 317
•
Reading only the context, not the steps (a, b, etc.), can you tell that the problem calls for a hypothesis test?
– Knowing that a test of hypothesis is called for, can you determine that a one-tail test is appropriate?
• Knowing that a one-tail test is to be used, can you set up the hypotheses?

9.4: t test of Hypothesis for the
Mean (σ Unknown)
• When σ is unknown, the distribution for x-bar is a “t” distribution with n-1 degrees of freedom.
• Use sample standard deviation “s” to estimate σ.
• This test is more commonly used than the z test.

Assumptions
• Random Sample.
• You must assume that the underlying population, i.e. the underlying random variable x is distributed normally.
• This test is very “robust” in that it does not lose power for small violations of the above assumption.

You still need your 5 step Hypothesis Test
Methodology.
– The critical value approach is the same as that for the z test.
– The p-value method does not work as well when done by hand because of limitations in the “t” table.
– One- and Two-tail tests are possible.
• You could add another step to check assumptions.

• 9.54, page 323

9.5: z Test of Hypothesis for the Proportion
• For the nominal variable—variable values are categories and you tend to describe the data set in terms of proportions.
• Both one- and two-tail tests are possible.
• Problem 9.72 on page 329 is a good example.

• The number of observations of interest (successes) and the number of uninteresting observations
(failures) are both at least 5.