copyright © 2012 by nelson education limited. chapter 7 hypothesis testing i: the one-sample case...

Copyright © 2012 by Nelson Education Limited.

Chapter 7Hypothesis Testing I: The One-Sample

Case

7-1


• The basic logic of hypothesis testing– Hypothesis testing for single sample means– The Five-Step Model

• Other material covered:– One- vs. Two- tailed tests– Type I vs. Type II error– Student’s t distribution– Hypothesis testing for single sample

proportions

In this presentation you will learn about:

7-2


• Hypothesis testing is designed to detect significant differences: differences that did not occur by random chance. Hypothesis testing is significance testing.

• This chapter focuses on the “one sample” case: we compare a random sample (from a large group) against a population.– Specifically, we compare a sample statistic to a

population parameter to see if there is a significant difference.

Hypothesis Testing

7-3


• The engineering department at a university has been accused of “grade inflation” so engineering majors have much higher GPAs than students in general.

Hypothesis Testing for Single Sample Means: An Example

7-4


• GPAs of all engineering majors should be compared with the GPAs of all students.–There are 1000s of engineering majors, far

too many to interview.

–How can the dispute be investigated without interviewing all engineering majors?

The value of the parameter, average GPA for all students, is 2.70 (μ = 2.70), with a standard deviation of 0.70 (σ = .70).


(continued)

7-5


• For a random sample of 117 engineering majors,

= 3.00 • There is a difference between the parameter (2.70)

and the statistic (3.00). – It seems that engineering majors do have higher GPAs.

• However, we are working with a random sample (not all engineering majors).

– The observed difference may have been caused by random chance.

X


(continued)

7-6


• Hence, there are two explanations, or hypotheses, for the difference:

1. The sample mean (3.00) is the same as the pop. mean (2.70). – The difference is trivial and caused by random

chance.

2. The difference is real (significant).– engineering majors are different from all students.


(continued)

7-7


• Formally, we can state the two hypotheses as:

Null Hypothesis (H0)

“The difference is caused by random chance.”

The H0 always states there is “no

significant difference.”


(continued)

7-8


OR

Alternative hypothesis (H1)

“The difference is real”.

The H1 always contradicts the H0.


(continued)

7-9


• One (and only one) of these explanations must be true. Which one?

– We assume the H0 is true.

• We ask, “What is the probability of getting the sample mean of 3.00 if

the H0 is true and all engineering majors really have a mean of 2.70?”


(continued)

7-10


• We use the Z score formula,

and Appendix A to determine the probability of getting the observed difference.


(continued)

7-11


–By convention, we use the .05 value as a guideline to identify differences that would be rare if H0 is true.

• If the probability is less than .05, the calculated or “obtained” Z score will be beyond ±1.96 or the “critical” Z score.


(continued)

7-12


• So, substituting the values into Formula 7.1,

we calculate a Z score of 4.6:

Z = = 4.611770

7203

...


(continued)

7-13


• Z (obtained) of +4.6 is beyond Z (critical) of ±1.96:

◦ A difference this large would be rare if H0 is true.

◦ We reject H0.

» This difference is significant.

» The GPA of engineering majors is significantly different from

the GPA of the general student body.


(continued)

7-14


• All the elements used in the example above can be formally organized into a five-step model:

1.Making assumptions and meeting test requirements.

2.Stating the null hypothesis.3.Selecting the sampling distribution and

establishing the critical region.4.Computing the test statistic.5.Making a decision and interpreting the results

of the test.

Hypothesis Testing: The Five Step Model

7-15


• Random sampling– Hypothesis testing assumes samples were

selected according to EPSEM. – The sample of 117 was randomly selected from

all engineering majors.

• Level of Measurement is Interval-Ratio (I-R)– GPA is I-R so the mean is an appropriate

statistic.

Step 1: Make Assumptions and Meet Test Requirements

7-16


• Sampling Distribution is normal in shape–This assumption is satisfied by using a large

sample (n>100). (See the Central Limit Theorem in Chapter 5).

Step 1: Make Assumptions and Meet Test Requirements (continued)

7-17


H0: μ = 2.7– The sample of 117 comes from a population

that has a GPA of 2.7. – The difference between 2.7 and 3.0 is trivial

and caused by random chance.

• H1: μ≠2.7– The sample of 117 comes a population that

does not have a GPA of 2.7. – The difference between 2.7 and 3.0 reflects

an actual difference between engineering majors and other students

Step 2: State the Null Hypothesis

7-18


• Sampling Distribution= Z– Alpha (α) = .05– α is the indicator of “rare” events.

– Any difference with a probability less than α is rare and will cause us to reject the H0.

• Critical Region begins at +1.96– This is the critical Z score associated with

α = .05, two-tailed test.

– If the obtained Z score falls in the C.R., reject the H0.

Step 3: Select Sampling Distribution and Establish the Critical Region

7-19


Z (obtained) = = 4.611770

7203

...

Step 4: Compute the Test Statistic

7-20


Step 5: Make Decision and Interpret Results

7-21


• The obtained Z score fell in the C.R., so we reject the H0.

–If the H0 were true, a sample outcome of 3.00 would be

unlikely.

–Therefore, the H0 is false and must be rejected.

• Engineering majors have a GPA that is significantly different from the general student body.

Step 5: Make Decision and Interpret Results (continued)

7-22


• In hypothesis testing, we try to identify statistically significant differences that did not occur by random chance.

– In this example, we rejected the H0 and concluded that the difference was significant.

• The difference between the parameter 2.70 and the statistic 3.00 was large and unlikely to have occurred by random chance.

• It is very likely that Engineering majors have GPAs higher than the general student body.

The Five Step Model: Summary

7-23


• Model is fairly rigid, but there are two crucial choices:

1. One-tailed or two-tailed test (Section 7.4)

2. Alpha (α) level (Section 7.5)

Crucial Choices in the Five Step Model

7-24


• Two-tailed: States that population mean is “not equal” to value stated in null hypothesis.

Example:

H1: μ≠2.7, where ≠ means “not equal to”

(Note, the GPA example illustrated above was a two-tailed test).

One- and Two-Tailed Hypothesis Tests

7-25


• One-tailed: Differences in a specific direction.

Example:

H1: μ>2.7, where > signifies “greater than”or

H1: μ<2.7, where < signifies “less than”

One- and Two-Tailed Hypothesis Tests (continued)

7-26


‣ Tails affect Critical Region in Step 3:

One- and Two-Tailed Hypothesis Tests (continued)

7-27


• By assigning an alpha level, α, one defines an “unlikely” sample outcome.

• Alpha level is the probability that the decision to reject the null hypothesis, H0, is incorrect. – Incorrectly rejecting a true null hypothesis:

Type I or alpha error

Alpha Levels

7-28


• Alpha levels affect Critical Region in Step 3:

Alpha Levels (continued)

7-29


• Type I, or alpha error: – Rejecting a true null hypothesis

• Type II, or beta error:– Failing to reject a false null hypothesis


7-30


• There is a relationship between decision making and error:


7-31


• How can we test a hypothesis when the population standard deviation (σ ) is not known, as is usually the case?–For large samples (n>100), we use s as an

estimator of σ and use standard normal distribution (Z scores), suitably corrected for the bias (n is replaced by n-1 to correct for the fact that s is a biased estimator of σ.

The Student’s t distribution

7-32


– For small samples (n<100), s is too unreliable an estimator of σ so do not use standard normal distribution. Instead we use Student’s t distribution.

The Student’s t distribution

7-33


A similar formula as for Z (obtained) is used in hypothesis testing:

The Student’s t distribution (continued)

7-34


The logic of the five-step model for hypothesis testing is followed.

However in testing hypothesis we use the t table (Appendix B), not the Z table (Appendix A).

The t table differs from the Z table in the following ways:

1.Column at left for degrees of freedom (df) (df = n – 1)

2.Alpha levels along top two rows: one- and two-tailed

3.Entries in table are actual scores: t (critical) (i.e., mark beginning of critical region, not areas under the curve).

The Student’s t distribution (continued)

7-35


Hypothesis Testing for Single-Sample Means: Summary

7-36


• We can also use Z (obtained) to test sample proportions, as long as sample size is large (n >100). However, the symbols change because we are basing the test on sample proportions (rather than sample means):

*Small-sample tests of hypothesis for proportions are not considered in this text.

Test of Hypothesis for Single-Sample Proportions (for Large Samples)*

7-37


where Ps = sample proportion

Pu = population proportion

= the standard deviation of the sampling distribution of

sample proportions

The logic of the five-step model for hypothesis testing is followed.

Test of Hypothesis for Single-Sample Proportions (for Large Samples) (continued)

7-38

copyright © 2012 by nelson education limited. chapter 7 hypothesis testing i: the one-sample case...

Documents