Modular 15 Ch 10.1 to 10.2 Part I. Ch 10.1 The Language of Hypothesis Testing Objective A : Set up a Hypothesis Testing Objective B : Type I or Type II.

Modular 15Ch 10.1 to 10.2 Part I

Ch 10.1 The Language of Hypothesis Testing

Objective A : Set up a Hypothesis Testing

Objective B : Type I or Type II Error

Ch 10.2 Hypothesis Test for a Population Mean

Objective C : State conclusions for Hypothesis Tests

Objective A : Classical Approach

Ch 10.1 The Language of Hypothesis TestingObjective A : Set up a Hypothesis Testing

Hypothesis testing is a procedure, based on a sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

– The null hypothesis is a statement to be tested. 0H

– The alternate hypothesis is a statement that we are trying to find evidence to support.

1H

Example 1 : Set up and .In the past, student average income was $6000 per year.

(a) An administrator believes the average income has increased.

(b) The percentage of passing a Math course was 50%. A Math professor believes there is a decrease in the passing rate.

(No change)6000:0 H

1H0H

(A right-tailed test)6000:1 H

5.0:0 pH

(A left-tailed test)5.0:1 pH

Ch 10.1 The Language of Hypothesis Testing

Objective A : Set up a Hypothesis Testing

Objective B : Type I or Type II Error

Ch 10.2 Hypothesis Test for a Population Mean

Objective C : State conclusions for Hypothesis Tests

Objective A : Classical Approach

Objective B : Type I or Type II Error

Type I Error →

Type II Error →

Rejecting when is true.0H 0H

We use for the probability of making Type I error.

We use for the probability of making Type II error.

For this statistics class, we only control the Type I error. )10.001.0(

is also called the level of significance.

Not rejecting when is true.0H 1H

Ch 10.1 The Language of Hypothesis Testing

Ch 10.1 The Language of Hypothesis Testing

Objective A : Set up a Hypothesis Testing

Objective B : Type I or Type II Error

Ch 10.2 Hypothesis Test for a Population Mean

Objective C : State conclusions for Hypothesis Tests

Objective A : Classical Approach

Objective C : State Conclusions for Hypothesis Tests

If is rejected, there is sufficient evidence to support the statement in .

0H1H

If is NOT rejected, there is NOT sufficient evidence to support the statement in .

0H1H

Section 10.1 The Language of Hypothesis Testing

Example 1 : In 2007, the mean SAT score on the reasoning test for all students was 710. A teacher believes that, due to the heavy use of multiple choice test questions, the mean SAT reasoning test has decreased.

0H(a) Determine and .

A Type I error is made when the sample evidence leads the teacher to believe is not true when in fact is true.

710

1H

710:0 H

710:1 H

(b) Explain what it would mean to make a Type I error.

710

A Type II error is made when the sample evidence leads the teacher to believe is true when in fact is less than 710.

710

710:0 H

710:1 H

(c) Explain what it would mean to make a Type II error.

(d) State the conclusion if the null hypothesis is rejected.

There is enough evidence to support the teacher’s claim that the mean SAT reasoning test score has decreased.

Example 2 : The mean score on the SAT Math Reasoning exam is 516. A test preparation company states that the mean score of

students who take its course is higher than 516.(a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the marketing campaign.

There is not sufficient evidence to support the mean score of the reasoning exam is higher than 516.

516:0 H

516:1 H

(b) If sample data indicate that the null hypothesis should not be rejected, state the conclusion of the company.

If we tested this hypothesis at the = 0.01 level, what is the probability of committing a Type I error?

(c) Suppose, in fact, that the mean score of students taking the preparatory course is 522. Has a Type I or Type II error been made?

Type II error.

(d) If we wanted to decrease the probability of making a Type II error, would we need to increase or decrease the level of significance?

If we increase , we decrease and vice versa.

Since is the probability of committing Type I error, it is 0.01.

Therefore, we need to increase the level of significance in order to decrease the probability of Type II error.

Example 3 : According to the Centers for Disease Control, 15.2% of adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches.(a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques.

There is sufficient evidence to support the massage therapist’s program that can reduce the percentage of migraine headache from 15.2%.

152.0:0 pH

152.0:1 pH

(b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program.

Type I error.

(c) Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is 15.3%. Was a Type I or Type II error committed?

Ch 10.1 The Language of Hypothesis Testing

Objective A : Set up a Hypothesis Testing

Objective B : Type I or Type II Error

Ch 10.2 Hypothesis Test for a Population Mean

Objective C : State conclusions for Hypothesis Tests

Objective A : Classical Approach

sample proportion, .

Ch 10.2 Hypothesis Tests for a Population Proportion

– Test for a Proportion A hypothesis test involving a population proportion can be considered as a binomial experiment. As we learned from Ch 8.2, the best point estimate of , the population proportion, is a

Z

p

n

xp ˆ

Objective A : Classical Approach

There are two methods for testing hypothesis.

Method 1 : The Classical Approach

Method 2 : The – Value ApproachP

We are going to focus on the Classical Approach for today’s lecture.

Testing Hypotheses Regarding a Population Proportion, .Use the following steps to perform a proportion – Test provided thatZ

p

Example 1: Use the Classical Approach to test the following hypotheses.versus25.0:0 pH 25.0:1 pH

25.0:0 pH

25.0:1 pH

Step 1 :

1.0,96,400 xn

Step 2 :

Step 3 :

1.0Sample : 96,400 xn

24.0400

96ˆ n

xp

npp

ppz

)1(

ˆ

00

00

400)25.01)(25.0(

25.024.0

400)75.0)(25.0(

25.024.00

z 46.0 * (test statistic)

(left-tailed test)

Use Table V to determine critical value for :1.0

*

From Table V

1003.0 0985.02.1

08.0

(Closer to 0.1)

46.00 z28.1z

1.0

Recall :

Since – statistic, , is not within the critical region, , fail to reject .

Z

25.0:1 pH (left-tailed test)

0H0z z

There is not enough evidence to support .25.0:1 pH

Step 4 :

Step 5 :

Example 2: The percentage of physicians who are women is 27.9%. In a survey of physicians employed by a large university health system, 45 of 120 randomly selected physicians were women. Use the Classical Approach to determine whether there is sufficient evidence at the 0.05 level of significance to conclude that the proportion of women physicians at the university health system exceeds 27.9%?

279.0:0 pH

279.0:1 pH

Step 1 :

Step 2 : 05.0

(right-tailed test)

Use Table V to determine critical value for :

Step 3 : Sample : 45,120 xn

375.0120

45ˆ n

xp

npp

ppz

)1(

ˆ

00

00

120)279.01)(279.0(

279.0375.0

120)721.0)(279.0(

279.0375.00

z 34.2 * (test statistic)

05.0

05.0

279.0:1 pH (right-tailed test)

From Table V

9495.0 9505.06.1

04.0

Since – statistic, ,falls within the critical region, , reject . Z 0H0z z

*

34.20 z645.1z

05.095.005.01 05.0

045.0

95.0

There is sufficient evidence to support the claim that the proportion of female physicians exceeds 27.9%.

Step 4 :

Step 5 :