Topic:One Tailed and Small Sample Hypothesis Testing

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Contents 1 Topic Highlights 2 Introduction and Motivation 3 Learning Activities 4 Learning Objectives 5 Topic Notes: One Tailed Hypothesis Testing 5.1 Concept 5.2 Example 1 5.3 Example 2 5.4 Example 3 5.5 Example 4 6 Topic Notes: The Hypothesis Testing Process - Updated 6.1 Hypothesis Testing Steps for Large Samples 6.2 Z Tables for Large Samples 6.2.1 Z table for Two-Tailed 6.2.2 Z table for One-Tailed 7 Topic Notes: Hypothesis Testing for Small Samples 7.1 Hypothesis Testing Steps for Small Samples 8 Practice Problems 8.1 Practice Problem 1 8.2 Practice Problem 2 8.3 Practice Problem 3

Topic Highlights

(What you will learn)

• That there is another basic type of hypothesis test (the one-tailed test)
• The difference betwee two and one-tailed testing
• Why you would use the one-tailed test
  
• How to use the one-tailed test
• How things are different for small samples

Introduction and Motivation

(Why learn it)

In the topic Two Tailed Hypothesis Testing, we looked at the basics of hypothesis testing in the case of "two tails", which usually applies to cases in which you want to demonstrate whether a claim about the population mean appears to be correct or not. In this topic, we are going to look at the idea of a one-tailed test. This kind of test usually applies to cases in which you want to demonstrate whether a population mean is larger or smaller than some specified or claimed value.

Then, we will take a step back and look at how things look for small samples, i.e. n<30; a practical reality in many cases.

Again, these types of hypothesis testing have many business applications that will be discussed.

Learning Activities

(How the levels of understanding will be gained)

Learning activities for this topic

Type	Name	Direction
Lecture and discussion	One Tailed Hypothesis Testing (below in this page) The Hypothesis Testing Process - Updated (also below) Hypothesis Testing for Small Samples (also below)	Instructor-directed
Reading	Section 8.1 and 8.3 of Bowerman et al. Or: Section 8.2 and 8.4 of Kvanli et al.	Self-directed
Personal activities	Review the learning objectives below	Self-directed

Learning Objectives

(Levels of understanding to be gained)

Learning objectives for this topic

Level of Understanding	Objective(s)
Very best	Can I immediately and consistantly recognize the difference between the cases requiring one-tailed and two-tailed hypothesis statements?
Highly satisfactory	Can I solve Practice Problem 3? Can I solve Practice Problem 2? Can I solve Practice Problem 1?
Satisfactory	Do I have the Z Tables for Large Samples on my equation sheet, and can I derive z for other values of ? Can I do the same for values of t?
Maybe just enough to pass	Do I understand the basic concept of one-tailed hypothesis testing, as described in Example 1, Example 2 and Example 3? Do I have the steps in the Hypothesis Testing Process - Updated on my equation sheet? Do I have the Hypothesis Testing Steps for Small Samples on my equation sheet?

Topic Notes: One Tailed Hypothesis Testing

We talked a lot about the concept of hypothesis testing in Two Tailed Hypothesis Testing, so here we're going to jump right in with how the one-tailed test is different.

Concept

In two-tailed hypothesis testing we considered examples where the manager wants to use statistics to evaluate whether the mean of a population appears to be equal to some value, based on a sample. The following hypothesis statement applied:

H₀: = ₀

Ha: ₀

Recall our example where a manager at a manufacturer of Compact Fluorescent Light (CFL) bulbs is challenged when its competitor claimed the average life time of its new product was 10000 hours. The challenge was to test whether the mean of the population was equal to 10000 hours.

From Two Tailed Hypothesis Testing, you will recall the following general sketch that applies for this situation:

This type of test is very useful in practice, but there are also cases where the manager may wish to demonstrate that the mean of a population is larger or smaller than some specified or claimed value.

Example 1

For example, the manager in the case of the CFL bulbs may not be satisfied only demonstrating whether the competitor's claim appears to be true. It may be more useful to him to demonstrate that the mean lifetime of the new bulbs actually appears to be less than 10000 hours.

In this case, the hypothesis statement would look something like the following:

H₀: = ₀ = 10000

Ha: < 10000

Here, the focus of our attention is on the alternative hypothesis, H_a. The null hypothesis, H₀, still contains the equal case (and the = sign), but the alternative hypothesis now contains the question in which we are interested in answering.  In this case, the following sketch applies:

Be clear that although we are interested in H_a, the goal of the hypothesis test is still to either reject H₀, or not reject H₀.

Example 2

I'm sure you can imagine that there are also cases where one wants to demonstrate that the population mean appears to be more than some value. The corresponding hypothesis statement would be:

H₀: = ₀ = 10000

Ha: > 10000

In this case, the following sketch applies:

Example 3

It is possible to write a one-tailed hypothesis statement in two ways. For example the following two statements are the same:

H0: = 0 Ha: > 0

H0: <= 0 Ha: > 0

Example 4

What is the z-score corresponding to each of the following significance levels in the one-tailed case?

(Hint: You'll need your table because they're different from their two-tailed equivalents.)

a) = 0.10

b) = 0.05

c) = 0.025

d) = 0.02

e) = 0.01

Solution
a) For =0.10, you need to look up the area 0.40: for which you should get z = 1.28. b) z = 1.645 c) z = 1.96 d) z = 2.055 e) z = 2.33

The last thing we need to do before moving on is to confirm that the test statistic is no different in the one-tailed case than it was for the two-tailed case:

Topic Notes: The Hypothesis Testing Process - Updated

Hypothesis Testing Steps for Large Samples

Now that you've seen the situation for the one-tailed case, let's summarize the basic steps for both cases:

1. Define the hypotheses, i.e.:

Two-tailed case	One-tailed case
H0: = 0 Ha: 0	H0: = 0 Ha: > 0      or: H0: = 0 Ha: < 0

2. Sketch the situation, e g.: (shown for = 0.10)

Two-tailed case	One-tailed case
	or:
Where the value of z you use is given in the Z Tables for Two-Tailed (below)	Where the value of z you use is given in the Z Tables for One-Tailed (below)

3. Compute the test statistic, i.e. using:

(for both cases)

4. Determine whether to reject

i.e. Provide a statement based on whether z* falls in the rejection region

5. Give a conclusion

e.g. Given the sample, n, and the level of significance, , ...

These steps should be on your equation sheet.

Z Tables for Large Samples

The following tables summarize the values of z for typical values of . The idea is that the columns in orange should be on your equation sheet. Those in grey show other relevant information and the intermediate steps. You should understand these intermediate steps; if you're given a significance value not found in this table, then you will need to look it up yourself.

Z table for Two-Tailed

Confidence level		/2	area to look up (=0.50 - /2)	z
90%	0.10	0.05	0.45	1.645
95%	0.05	0.025	0.475	1.96
97.5%	0.025	0.0125	0.4875	2.24
98%	0.02	0.01	0.49	2.33
99%	0.01	0.005	0.495	2.575

Z table for One-Tailed

	area to look up (=0.50 - )	z
0.10	0.40	1.28
0.05	0.45	1.645
0.025	0.475	1.96
0.02	0.48	2.055
0.01	0.49	2.33

Topic Notes: Hypothesis Testing for Small Samples

All of the examples we've considered so far have been for cases where we've taken a sample with more than 30 data points. You should recall from the topic More on Confidence Intervals that when n < 30 we used the t distribution instead of the standard normal distribution.

We do the same thing for hypothesis testing: when n < 30, use the t distribution.

In practice, things are very similar to what was summarized above for large samples, except that:

1. You need to look up t instead of z
2. To do this, you also need degrees of freedom, df = n-1

The following sketch applies for the two-tailed case:

We can't create a single table of t-values because they depend on the degrees of freedom. You should recall the following however:

Confidence level		Associated right-side area (/2) for the t table	value of t
80%	0.20	0.10	need to look up using df
90%	0.10	0.05	"
95%	0.05	0.025	"
98%	0.02	0.01	"
99%	0.01	0.005	"

The following sketches apply for the one-tailed case:

And you should understand the following:

	Associated right side area () for the table	value of t
0.10	0.10	need to look up using df
0.05	0.05	"
0.025	0.025	"
0.01	0.01	"
0.005	0.005	"

Once you understand the above situation, the hypothesis testing steps are the same for the small sample case.

Hypothesis Testing Steps for Small Samples

Now that you've seen the situation for small samples, let's summarize the basic steps for both cases (large and small):

1. Define the hypotheses - no difference from large sample, i.e.:

Two-tailed case	One-tailed case
H0: = 0 Ha: 0	H0: = 0 Ha: > 0      or: H0: = 0 Ha: < 0

2. Sketch the situation, e g.: (shown for = 0.10, n = 11)

Two-tailed case	One-tailed case
	or:
Where you look up the value of t for /2 and df = n-1	Where you look up the value of t for and df = n-1

3. Compute the test statistic, i.e. using:

 (for both cases)

4. Determine whether to reject

i.e. Provide a statement based on whether t* falls in the rejection region

5. Give a conclusion

e.g. Given the sample, n, and the level of significance, , ...

These steps should be on your equation sheet, although you may want to combine them somehow with the steps given earlier for the large sample case, since they are so similar in structure.

Practice Problems

Practice Problem 1

To be handed out in class.

Practice Problem 2

To be handed out in class.

Practice Problem 3

To be handed out in class.