Section 9.1

Introduction to Statistical Tests

9.1 / 1

Hypothesis testing is used to make decisions concerning the value of a parameter.

Null Hypothesis: H0

• Is a working hypothesis about the population parameter in question

• The value specified in the null hypothesis is often:

• a historical value• a claim• a production specification

9.1 / 2

Alternate Hypothesis: H1

• Is any hypothesis that differs from the null hypothesis

• An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.

9.1 / 3

A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.

9.1 / 4

Example

The null hypothesis (the claim) is that the true average life is 1000 hours. H0: μ = 1000

If we reject the manufacturer’s claim, we must accept the alternate hypothesis that the light bulbs do not last as long as 1000 hours. H1: μ < 1000

Types of Statistical Tests

• Left-tailed: H1 states that the parameter is less than the value claimed in H0.

• Right-tailed: H1 states that the parameter is greater than the value claimed in H0.

• Two-tailed: H1 states that the parameter is different from ( ) the value claimed in H0.

9.1 / 5

Given the Null Hypothesis H0: = k

If you believe that is less than k,Use the left-tailed test: H1: < k

If you believe that is more than k,Use the right-tailed test: H1: > k

If you believe that is different from k,Use the two-tailed test: H1: k

General Procedure for Hypothesis Testing

• Formulate the null and alternate hypotheses.• Take a simple random sample.• Compute a test statistic corresponding to the

parameter in H0.• Assess the compatibility of the test statistic

with H0.

9.1 / 7

8

Hypothesis Testing about the Mean of a Normal Distribution with

a Known Standard Deviation

x-test statistic z

/ n

size sample n

H in stated value

sample random simple of meanx

0

9.1 / 9

ExampleStatistical Testing Preview

Page 364

P-value of a Statistical Test

• Assuming H0 is true, the probability that the test statistic (computed from sample data) will take on values as extreme as or more than the observed test statistic is called the P-value of the test

• The smaller the P-value computed from sample data, the stronger the evidence against H0.

9.1 / 10

P-values for Testing a Mean Using the Standard Normal Distribution

n/

-x z

statistictest sample edstandardiz

the Use

x

9.1 / 11

P-value for a Left-tailed Test

• P-value = probability of getting a test statistic less than xz

9.1 / 12

• P-value = probability of getting a test statistic greater than xz

P-value for a Right-tailed Test

9.1 / 13

• P-value = probability of getting a test statistic lower than or higher thanxz xz

P-value for a Two-tailed Test

9.1 / 14

Types of Errors in Hypothesis TestingType I Errorrejecting a null hypothesis which is, in fact, trueType II Errornot rejecting a null hypothesis which is, in fact, false

Type I and Type II Errors

9.1 / 15

Types of ErrorsFor tests of hypotheses to be well constructed, they must be

designed to minimize possible errors of decision. (Usually we don’t know if an error has been made, and therefore, we can talk only about the probability of making an error.)

Usually, for a given size, an attempt to reduce the probability of one type of error results in an increase in the probability of the other type of error.

In practical applications, one type of error may be more serious than the other. In such case, careful attention is given to the more serious error. If we increase the sample sizes, it is possible to reduce both types of errors, but increasing the sample size may not be possible.

9.1 / 16

Types of Errors

Good statistical practice requires that we announce in advance how much evidence against will be required to reject .

The probability with which we are willing to risk a type of I error is called the level of significance of a test. (Reject a true )

The level of significance is denoted by the Greek letter a (pronounced “alpha”).

9.1 / 17

0H

0H

0H

0H

Level of Significance, Alpha (a)the probability of rejecting a true hypothesisAlpha = a is the probability of a type I error

Type II Error

Beta = β = probability of a type II error (failing to reject a false hypothesis)

In hypothesis testing α and β values should be chosen as small as possible.

Usually α is chosen first.9.1 / 18

Power of the Test = 1 – β

Is the probability of rejecting H0 when it is in fact false = 1 – b.

The power of the test increases as the level of significance (a) increases.

Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

9.1 / 19

Probabilities Associated with a Statistical Test

9.1 / 20

ExampleHypotheses and Types of Errors

A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. We wish to test the claim with a level of significance of a = 0.01.

Determine the Null and Alternate hypotheses and describe Type I and Type II errors.

9.1 / 21

… average age of its job applicants is fifteen years. We suspect that the true age is lower than 15.

9.1 / 22

H0: m = 15 H1: m < 15

A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

a = 0.01

A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.

Concluding a Hypothesis Test Using the P-value and Level of Significance α

• If P-value < α reject the null hypothesis and say that the data are statistically significant at the level α.

• If P-value > α, do not reject the null hypothesis.

9.1 / 23

Basic Components of a Statistical Test

1. Null hypothesis, alternate hypothesis and level of significance

2. Test statistic and sampling distribution3. P-value4. Test conclusion5. Interpretation of the test results

9.1 / 24

1. Null Hypothesis, Alternate Hypothesis and Level of Significance

If the sample data evidence against H0 is strong enough, we reject H0 and adopt H1.

The level of significance, α, is the probability of rejecting H0 when it is in fact true.

9.1 / 25

2. Test Statistic and Sampling Distribution

Mathematical tools to measure compatibility of sample data and the null hypothesis

3. P-valueThe probability of obtaining a test statistic from the

sampling distribution that is as extreme as or more extreme than the sample test statistic computed from the data under the assumption that H0 is true

9.1 / 26

4. Test ConclusionIf P-value < α reject the null hypothesis and say that the data are statistically significant at the level α.If P-value > α, do not reject the null hypothesis.

5. Interpretation of Test ResultsGive a simple explanation of conclusion in the context of the application.

Example

Guided exercise 3 page 370

9.1 / 27

Reject or ...

• When the sample evidence is not strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis.

• Use of the term “accept the null hypothesis” should be avoided.

• When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

9.1 / 28

Fail to Reject H0

• There is not enough evidence to reject H0. The null hypothesis is retained but not proved.

9.1 / 29

Reject H0

• There is enough evidence to reject H0. Choose the alternate hypothesis with the understanding that it has not been proven.

9.1 / 30