HYPOTHESIS TESTING

Dadan Kusnandar, Ph.D.

1

OBJECTIVES

When you have completed this topic, you should be able to:

• Develop the hypothesis testing procedure as a technique for decision making
• Determine the risks involved in making these decisions based only upon sample information

2

CONCEPTUAL EMPHASIS

1. Basic concepts of hypothesis testing
2. One sample tests for means, proportions and variance
3. Two sample tests for a mean
4. The paired difference test

3

Readings

• Berenson and Levine (1992) Chapters 11, 12 & 13.
• Kusnandar et al (2009) Chapter 6.

CONCEPTS OF HYPOTHESIS TESTING

The objective of hypothesis testing is to determine whether or not the sample data support some belief or hypothesis about the population

Examples:

• A company that has a 10% market share launches a new advertising campaign. At the campaign’s completion, the company wants to know whether the results of a random sample indicate an increase in market share
• A firm that produce fertilizer want s to know whether a new type of fertilizer increases crop yields.

4

STRUCTURE OF HYPOTHESIS TESTING

• Null hypothesis
• Alternative hypothesis
• Test statistic
• Rejection region (critical region)

5

THE NULL HYPOTHESIS (H₀) AND �THE ALTERNATIVE HYPOTHESIS (H₁)

The purpose of hypothesis testing is to choose between two conflicting hypotheses about the possible value of a population parameter

The null hypothesis (H₀) is an assumption concerning the value of the population parameter.

The alternative hypothesis (H₁) specifies all possible values of the population parameter that are not specify in the null hypothesis

​

6

CRUCIAL THINGS ABOUT THE TWO HYPOTHESES

• The null hypothesis always specify that the parameter is equal to a single value

H₀: θ = θ₀

• The alternative hypothesis can be expressed in one of the following:
• H₁: θ ≠ θ₀ → two sided alternative hypothesis
• H₁: θ < θ₀ → one sided alternative hypothesis
• H₁: θ > θ₀ → one sided alternative hypothesis

7

TEST STATISTIC AND REJECTION REGION

• Test statistic is the criterion upon which we base our decision whether to reject or not to reject the null hypothesis
• The test statistic is the point estimator of the parameter being tested
• The rejection region is a range of values such that, if the test statistic falls into that range, we decide to reject the null hypothesis

8

REJECTION REGION FOR�H₀: θ = θ₀ VS. H₁: θ ≠ θ₀

9

REJECTION REGION FOR�H₀: θ = θ₀ VS. H₁: θ > θ₀

10

REJECTION REGION FOR�H₀: θ = θ₀ VS. H₁: θ < θ₀

11

TYPE I AND TYPE II ERRORS

• Type I error occurs if the null hypothesis is rejected when in fact it is true.�The probability of a Type I error is denoted by α, called as the level of significance
• Type II error occurs if the null hypothesis is not rejected when it is false.�The probability of a Type II error is denoted by β. The complement (1 – β) is called the power of a statistical test

12

RELATIONSHIP BETWEEN α AND β

13

PROCEDURE OF HYPOTHESIS TESTING

1. Specify the null hypothesis and the alternative hypothesis
2. Specify the test statistic
3. Specify α, and set up the rejection region
4. Calculate the value of the test statistic
5. Draw the conclusion: reject or do not reject H₀

14

TEST OF HYPOTHESIS FOR THE MEAN, Σ_x KNOWN

When the sample size n is large, the Central Limit Theorem states that the sampling distribution of the sample mean would follow the Normal Distribution with mean μ and variance σ²/n. Hence the test statistic would be as follows:

15

For a specified level of significance (α), the rejection region can then be determined by utilizing the Standardized Normal Distribution

REJECTION RULES

16

Test statistic:

EXAMPLE 1�BERENSON & LEVINE (1992) PROBLEMS 11.13 P.365

Suppose that the director of manufacturing at clothing factory needed to determine whether a new machine was producing a particular type of clothing according to the manufacturer’s specification, which indicate that the cloth should have a mean breaking strength of 70 pound and a standard deviation of 3.5 pounds. A sample of 36 pieces revealed a sample mean of 69.7 pounds. Is there evidence that the machine is not meeting the manufacturer’s specification in term of the average breaking strength? (Use the .05 level of significance)

17

SOLUTION TO EXAMPLE 1

1. The null and alternative hypothesis were

H₀: μ = 70

H₁: μ ≠ 70

• The test statistic would be the Z statistic
• Significance level of .05 specified the size of the rejection region. Since H₁ is a two sided hypothesis, the rejection region is divided into the two tails of the distribution, i.e. 0.025 each. Looking up the area in the Normal distribution the critical value that divide the rejection and nonrejection region are +1.96 and –1.96��The rejection rule would be�Reject H₀ if Z > +1.96 or Z < – 1.96�Do not reject H₀ if –1.96 ≤ Z ≤ +1.96

18

SOLUTION TO EXAMPLE 1 (CONTINUED)

4. The value of the test statistics were

​

​

​

​

​

• Since Z = – 0.514, our decision is not to reject H₀ �(we see that –1.96 < – 0.514 < +1.96)

USING MINITAB

• Store the data in one column of Minitab spreadsheet
• Choose Stats ⮞ Basic Statistics ⮞ 1-Sample z
• Click Test mean button and complete the dialog box

20

MINITAB OUTPUT

21

Z-Test

​

Test of mu = 70.000 vs mu not = 70.000

The assumed sigma = 3.50

​

Variable N Mean StDev SE Mean Z P

strength 36 69.700 3.452 0.583 -0.51 0.61

H₀: μ = 70

H₁: μ ≠ 70

The value of Z statistic

The p value: the probability of obtaining a test statistic equal to or more extreme than the result observed, given H₀ is true

TEST OF HYPOTHESIS FOR THE MEAN, Σ_X UNKNOWN

22

Test statistic:

EXAMPLE 2

Suppose that the standard deviation of the population in Example 1 was unknown. However, we have the sample standard deviation, which is 3.452 pounds. Is there any evidence that the machine is not meeting the manufacturer’s specification ?

23

MINITAB OUTPUT

24

Stats ⮞ Basic Statistics ⮞ 1-Sample t

T-Test of the Mean

​

Test of mu = 70.000 vs mu not = 70.000

​

Variable N Mean StDev SE Mean T P

strength 36 69.700 3.452 0.575 -0.52 0.61

RULE OF THUMB

25

Test of hypothesis for the mean

TEST OF HYPOTHESIS FOR A PROPORTION

26

Test statistic:

EXAMPLE 3

A stationery supply store receives a shipment of a certain brand of inexpensive ball point pens from the manufacturer. The shipment can be returned if there is more than 5% defective. A random sample of 300 pens is tested and 30 are found to be defective. Can the owner returned this shipment? Use a level of significance of .10.

27

THE CHI-SQUARE DISTRIBUTION

28

The statistic:

will follow the Chi-square distribution with (n – 1) degrees of freedom

CRITICAL VALUES OF Χ²

29

TESTING A HYPOTHESIS ABOUT A POPULATION VARIANCE

30

The statistic:

EXAMPLE 4

A manufacture of candy must monitor the temperature at which the candies are baked. Too much variation will cause inconsistency in the taste of the candy. Past records show that the standard deviation of the temperature has been 1.2^o F. A random sample of 30 batches of candy is selected and the sample standard deviation of the temperature is 2.1^o F. Is there any evidence that the population standard deviation has increase above 1.2^o F?

31

TWO-SAMPLE TESTS

32

?

!

TESTING FOR DIFFERENCE BETWEEN THE MEANS OF TWO INDEPENDENT POPULATIONS

• Two-tailed test:

H₀: μ₁ = μ₂ or μ₁ – μ₂ = 0

H₁: μ₁ ≠ μ₂ or μ₁ – μ₂ ≠ 0

• One-tailed tests:

H₀: μ₁ = μ₂ or μ₁ – μ₂ = 0

H₁: μ₁ > μ₂ or μ₁ – μ₂ > 0

or

H₀: μ₁ = μ₂ or μ₁ – μ₂ = 0

H₁: μ₁ < μ₂ or μ₁ – μ₂ < 0

33

Case 1. Both population variances are known, or both sample sizes > 30

34

The test statistic is

or

For a specified level of significance (α), the rejection region can then be determined by utilizing the Standardized Normal Distribution

EXAMPLE 5

Management of the Sycamore Steel Co. wishes to determine if there is any difference in performance between the day shift of workers and the evening shift of workers. A sample of 120 day-shift workers reveals an average output of 74.3 parts per hour with a standard deviation of 16 parts per hour. A sample of 100 evening-shift workers reveals an average output of 69.7 parts per hour with a standard deviation of 18 parts per hour. Is there any evidence of a difference in output between the day shift and evening shift?

35

Case 2. Small sample sizes: both population variances are unknown, but can be assumed to be equal

36

The test statistic is

where

For a specified level of significance (α), the rejection region can then be determined by utilizing the t-distribution with (n₁+n₂ – 2) degrees of freedom

EXAMPLE 6

A quality control manager at a light bulb factory would like to determine if there is any difference in the average life of bulb manufactured on two different types of machines. A random sample of 25 light bulbs obtained from machine 1 indicated a sample mean of 375 hours with a sample standard deviation of 110 hours, and a similar sample of 25 from machine 2 indicated a sample mean of 362 hours with a sample standard deviation of 125 hours. Is there any evidence of a difference in the average life of bulbs produced by the two types of machines?

37

Case 3. Small sample sizes: both population variances are unknown, and cannot be assumed to be equal

38

The test statistic is

For a specified level of significance (α), the rejection region can then be determined by utilizing an approximation to a t-distribution with n degrees of freedom, where

EXAMPLE 7

An experiment was conducted to compare the mean number of tapeworm in the stomachs of sheep that have been treated for worms against the mean number in those that were untreated. A sample of 14 worm-infected lambs was randomly divided into two groups. Seven were injected with the drug and the remainder were left untreated. After a six-month period only 13 animals were available for analysis and the following worm counts were recorded:

39

Is there any difference in the mean number of worms between treated and untreated lambs?

TESTING FOR DIFFERENCE BETWEEN THE MEANS FROM TWO RELATED POPULATIONS

• Two samples are related when the items or individuals in the first sample are not independent of the second sample
• The dependency occurs either because the items are paired or matched according to some characteristics or because repeated measurements are obtained from the same set of items

40

The variable of interest is the difference between the values of the observations rather than the observations themselves

41

42

Test statistic:

The paired difference test

where

EXAMPLE 8

The marketing manager of a company decided to evaluate the effectiveness of a new advertising campaign. He collected data on monthly sales before and during the campaign for eight regional stores (see the table). What should he conclude?

43