1

Where we are going: a graphic: Hypothesis Testing.

1 2 Paired 2 or more

Means

Variances

Proportions

Categories

Slopes

Ho: / CI

Ho: / CI

Samples

Ho: / CI Ho: / CI

Ho: / CI

Ho: / CI

Ho: / CI

Ho: Ho:

Ho: / CI

Ho: / CI

2

One Sample: Hypothesis testing.

population sample

Gather data

Make inferences and comparisons

parameters statistics

.,,,,, 2 etc

.etc,b,p,SS,or

.,ˆ,ˆ,ˆ,ˆ,ˆ2

2

y

etc

3

Hypothesis testing: 8 steps (briefly):

1. Research hypothesis (Ha:).2. Null hypothesis (Ho:).3. A. State type I error. B. State type II error.4. State , (the probability of type I error).5. Draw the rejection region; state rejection criterion.6. Compute the test statistic.7. Make a decision about Ho:.8. State a conclusion.

These will be explained in the context of examples.

4

An hypothesis about the mean. (Pulse rate example).

Consider that we are interested in knowing if average 15second pulse rate is different than 20. It has been shown that instatistics classes, if average pulse rate is different from 20,a large portion of students are thinking about something elseand retention of material is low. At this point, students needto be jolted somehow to regain consciousness. Is a jolt needed ?

1. Research (alternate) hypothesis: Ha: ≠ 20.

2. Null hypotheses: Ho: = 20.

3. A. Type I error: Claim pulse rate is different than 20 when in fact it is not (reject Ho|Ho is true).

B. Type II error: Claim pulse rate is equal to 20when in fact it is not (accept Ho|Ho is false).

5

An hypothesis about the mean, continued, 1.

4. Type I error tolerated: let = .05.

5. Rejection Region: .025 .025

t/2 ,(n-1) = t.025,127 = -1.96 1.96

Reject Ho if tobs is less than -1.96.

or if tobs is greater than 1.96.

6. Test statistic: nSy

tobs0

6

An hypothesis about the mean, continued, 2.

At this point, an experiment is conducted and data are collected. The following are the results:

128 measurements on 15 second pulse rate were made. the mean was 19.60 and the standard deviation was 2.24.

So: 6.

7. Decision: Since -2.02 < -1.96 , Reject Ho: (plot -2.02 on the drawing in 5) in favor of Ha:.

02.2198.

40.0

12824.2

2060.19tc

7

An hypothesis about the mean, continued, 3.

8. Conclusion. A recent study of 128 randomlyselected students showed that average 15 second pulse rate is lower than 20. A joltback to consciousness seems to be necessary. What would you suggest ?

We are 95% certain our answer is correct; or, there is a 5% chance we have rejected incorrectly. If the null is in fact correct,5 out of 100 studies done in this way will lead to rejection.

8

Three kinds of research/null hypotheses:

Rejection region for t: I. Ha: < o

Ho: o 0

II. Ha: > o Ho: o 0

III. Ha: o /2 /2 Ho: = o

0