CHAPTER 8 INTRODUCTION TO HYPOTHESIS TESTINGILLI NURASHIKIN BT MOHD ISA (GS38619)NAQIAH BT PUAAD (GS38686)NUUR ADILA MOHAMAD ALI (GS)

Subtopics 8.1 The Logic of Hypothesis Testing8.2 Uncertainty and Errors in Hypothesis Testing8.3 More About Hypothesis Testing8.4 Directional (One-Tailed) Hypothesis Tests8.5 Concerns About Hypothesis Testing: Measuring Effect Size8.6 Statistical Power

8.1 Logic of Hypothesis TestingHypothesis testing A statistical method that uses sample data to evaluate a hypothesis about a population.It is one of the most commonly used inferential procedures. Hypothesis tests is used to evaluate the results of the study.

Example:Purpose of research is to determine the effect of a treatment on an individuals in the population. Assuming = 80 and =20.

It is impossible to administer the treatment on the entire population, so the actual research study is conducted using a sample.Figure above basically shows the structure of the research study from the point of view of the hypothesis test.

Four steps of hypothesis testSTEP 1: State the hypothesisSTEP 2: Set the criteria for a decisionSTEP 3: Collect data and compute sample statisticsSTEP 4: Make a decision

Example 8.1 (pg 206)For general population, Mean, = 80Standard deviation, = 20Study on sample, n = 25 If the mean score is different from the mean for general population of students, researcher conclude that electric stimulation have an effect on mathematical skill.

Step 1: State the hypothesis

Null hypothesis (H)Alternative hypothesis (H1 )States that in the general population, there is no change, no difference, or no relationship. States that there is a change, a difference, or a relationship for the general population.Based on the experiment, H predicts that the independent variable (treatment) has no effects on the dependent variable (scores) for the population. Based on the experiment, H predicts that the independent variable (treatment) does have an effect on the dependent variable. H : (with stimulation) = 80

Even with the stimulation, the mean test score is still 80.H : ( with stimulation) 80

With the stimulation, the mean test score is different from 80.

Step 2: Set the criteria for a decisionThe alpha level or the level of significance, is a probability value that is used to define the concept of very unlikely in a hypothesis test. alpha values commonly used are: = .05 (5%) = .01 (1%) = .001 (0.1%)

Critical region is composed of the extreme sample values that are very unlikely (as defined by the alpha level) to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected.

Step 3: Collect data and compute sample statisticsComparing the data with the hypothesisCalculate a z-score that identifies where our sample mean is located in the hypothesized distribution. z = M - M z = sample mean hypothesized population mean standard error between M and

Step 4: Make a decisionResearcher uses the z-score value obtain in step 3 to make a decision about the null hypothesis according to the criteria established in step 2. There are 2 possible outcomes.

Sample data are located in the critical region.Sample data are not in the critical region.Reject the null hypothesisFail to reject the null hypothesis.Example: Sample mean, M = 89,Population mean, = 80,n = 25, and = 20. Standard error for the sample mean is : M = = 20 = 20 = 4 n 25 5 z = M - = 89-80 = 9 = 2.25 M 4 4alpha level of = .05, the z-score beyond the boundary of 1.96. Example: Sample mean, M = 84,Population mean, = 80,n = 25, and = 20. Standard error for the sample mean is : M = = 20 = 20 = 4 n 25 5 z = M - = 84-80 = 4 = 1.00 M 4 4alpha level of = .05, the z-core is in the boundary of 1.96.

8.2 Uncertainty and Errors In Hypothesis TestingTYPE I ERRORSOccurs when a researcher rejects a null hypothesis that is actually true. In a typical research situation, a Type I error means that the researcher concludes that a treatment does have an effect when in fact, it has no effect.It occurs when a researcher unknowingly obtains an extreme, non-representative sample. The alpha level determines the probability of a Type I error.How to avoid Type I error?Use a lower value for . However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists.

TYPE II ERRORSOccurs when a researcher fails to reject a null hypothesis that is really false. In a typical research situation, a Type II error means that the hypothesis test has failed to detect a real treatment effect. The research data do not show the results that the researcher had hoped to obtain. It occurs when the sample mean is not in the critical region even though the treatment has had an effect on the sample. (happens when the effect of the treatment is relatively small) How to avoid Type II error? By ensuring the sample size is large enough to detect a practical difference when one truly exists.

Selecting an alpha levelAlpha level Helps determine the boundaries for critical regionDetermine the probability of a Type I error if the null hypothesis is truePrimary concern when selecting an alpha level is to minimize the risk of a Type I error.Best strategy is to choose the smallest possible value to minimize the risk of a Type I error.

8.3 More About Hypothesis TestingIn the literature

A result said to be significant when the result is sufficient to reject the null hypothesis. Thus, a treatment has a significant effect if the decision from the hypothesis test is to reject Ho.The z indicates the z score used to evaluate the sample data and that is the value is 2.25p

Figure 8.6Sample means that fall in the critical region (shaded areas) have probability less than alpha (p).p>Fail to reject Hop

Assumptions for Hypothesis tests with z-scoresRandom SamplingParticipants used in the study were selected randomly. To generalize the findings from the sample to the population, the sample must be representative of the population. Random sampling helps to ensure that it is representative.

The value of is unchanged by the treatment.A critical part of the z-score formula in hypothesis test is the standard error, M. To compute the value for the standard error, must know the sample size (n) and the population standard deviation ().

Normal sampling distributionTo evaluate hypothesis with z-scores we have used the unit normal table to identify the critical region. This table can be used only if the distribution of sample means is normal

A closer look at the z-score in a hypothesis test

The z-score formula forms a ratio

For example, z-score of z =3.00 means that obtained difference between the sample and the hypothesis is 3 times bigger than would be expected if the treatment had no effect. A discrepancy this large is strong indication that the hypothesis is probably wrong.

Factors that influence a hypothesis testThe final decision in a hypothesis test determined by the value obtained for the z-score statistics. If the z-score is large enough to be in critical region, then we reject the null hypothesis and conclude there is significant treatment effect and vice versa.The variability of the scoresHigher variability can reduce the chances of finding a significant treatment effect. Example: if the standard deviation is increased to = 30. with the increased variability, the standard error becomes M = 30 = 6 points 25

The new z-score becomes89-80 = 9/6 = 1.50 6

The z-score is no longer beyond the critical boundary of 1.96 so the statistical decision is to fail to reject the null hypothesis. In general increasing the variability of the scores produces a large standard error and a smaller value (closer to zero) for the z-score.

2. The number of scores in the sample

In the example 8.1 a significant z-score of z= 2.25. If the sample size to n=100 students. With n=100, the standard error becomes M = 20 =2 100The z-score becomes 89-80 = 9/2= 4.50 2Increasing the no. of scores in the sample produces smaller standard error and larger value for the z-score.

Learning check

A researcher conducts a hypothesis test with = .05 to evaluate the effectiveness of a treatment. Assume that the sample mean produces a z-score of z = 2.17a) do the data indicate that the treatment has a significant effect?Answer: with =.05 the critical region consists of z-scores in the tails beyond z=1.96. Reject the null hypothesis

2. In research report the result hypothesis include the phrase z=1.63 p>.05. This means that the test failed reject the null hypothesis? (T/F)Answer: True

If all other factors are held constant, the larger the sample size, the greater the likelihood of finding a significant treatment effect.

8.4 Directional (One-tailed) Hypothesis Tests In a directional hypothesis test, or a one-tailed test the statistical hypotheses (Ho and H1) specify either an increase or a decrease in the population mean. That is, they make a statement about the direction of the effectExample 8.2 (pg.224)

The Hypothesis for a Directional TestBecause a specific direction is expected for the treatment effect, it is possible for the researcher to perform directional test.The first step is to incorporate the directional prediction into the statement of the statistical hypotheses.To express directional hypotheses in symbols, it usually begin with the alternative hypothesis (H1).

H1: > 80 (with the stimulation, the average score is greater than 80)The null hypothesis states that the stimulation does not increase scores,Ho: 80 (with the stimulation the average score is not greater than 80)

The Critical Region for Directional TestsCritical region is located entirely in the right-hand tail of the distribution corresponding to sample mean much greater than = 80. Because the critical region is contained in one tail-distribution, a directional test commonly called one-tailed test. =80M = 401.65MzReject Ho

The alpha level is not divided between two tails but rather contained entirely in one tail.Using = .05, the whole 5% is located in one tail. The score boundary for critical region is z= 1.65Directional (one-test) test requires changes the first 2 steps of the steps hypothesis testing procedure. First step, the directional prediction is included in the statements of the hypotheses

Second step, the critical region located entirely in one tail of the distribution.

For this example, the researcher obtained a mean of M=87 for the 25 participants who received the brain stimulation, so sample mean corresponds to a z-score 87-80 = 7/4= 1.75 4A z-score of z=1.75 in the critical region for one tailed test. Therefore, we reject the null hypothesis and conclude that the electrical stimulation produces significant increase in mathematics test scores.

In the literature, this result would be reported as follows:The stimulation produced significant increase in scores, z = 1.75, p

The major distinction between one tailed and two-tailed test is the criteria that they use for rejecting Ho .One tailed test allows you to reject the null hypothesis when the differences between the sample and population is relatively small, provided that the difference is in the specified direction.Two-tailed test requires a relatively large difference independent of direction.Comparison of One Tailed versus Two-Tailed Tests

Example 8.3 (pg. 226)

With the two-tailed test with the 7 point difference between sample mean and hypothesized population mean (M=87 and =80) is not big enough to reject null hypothesis.However, with one tailed test, the same 7 point difference is large enough to reject Ho and conclude that has significant effect.

Learning check

A researcher selects a sample from population with mean of = 60 and administers a treatment to the individuals in the sample. If the researcher predicts that the treatment will increase scores, thena) Using symbols, state the hypothesis for one tailed test.Answer: Ho : 60 and H1: > 60

2. A researcher obtains z= 2.43 for hypothesis test. Using =.01 the researcher should reject the null hypothesis for one-tailed test but fail to reject for two-tailed test. (T/F)Answer: True. The one-tailed critical value is z = 2.33 and the two-tailed value z is = 2.58.

8.5 Concerns about hypothesis testing : measuring effect size Two limitations to establish the significance of treatment effect

1. the focus of the hypothesis test is on the data rather than the hypothesis.2. a significant treatment effect does not necessarily indicate a substantial treatment effect.

Measuring effect sizeTo provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used

Standard deviation is included to standardize the size of the mean difference in much the way that z-scores standardize locations in the distribution.

Example:

Part (a) shows the results of the treatment that produces a 15-point mean difference in SAT scores; before treatment the average SAT score is 500, after treatment the average score is 515. Notice the standard deviation for SAT score is 100, so the 15-point difference appears to be small

Cohens d = mean difference = 515-500 = 0.15standard deviation 100

Part (b), treatment produces 15-point mean difference in IQ score. Before treatment the average IQ is 100, and after treatment the average is 115. because IQ score have standard deviation of 15, 15-point mean difference now appears to be large.

Cohens d = mean difference = 115-100 = 1.00standard deviation 15

8.6 Statistical powerAnother alternative approach to measure the size or strength of the treatment effectThe probability that the test will correctly reject a false null hypothesis.

Example :PLEASE FIX THE PICTURE

Normal-shaped population with a mean of 80 and standard deviation of 10. a researcher plans to select a sample of 25 individuals from this population and administer a treatment to each individual. It is expected that the treatment will have an 8-point effect, that is the treatment will add 8 points to each individual score.

Power and effect sizeAs the effect size increases, the distribution of sample means on right-hand size moves even farther to the right so that more and more of the samples are beyond the z boundary. As the effect size increase, probability of rejecting H0 also increase, which means the power of the test increases

Other factors that effect powerSample size larger sample produces greater power for the hypothesis testAlpha level reducing alpha level for a hypothesis test also reduces the power of a testChanging from a regular two-tailed test to one-tailed test increases the power of the hypothesis test

Example :PLEASE FIX THE PICTURE

Thank You

After the formula explain about it.*