Hypothesis Tests l Chapter 7 l 7.1 Developing Null and Alternative Hypotheses 7.2 Type I & Type II Error 7.3 Population mean 7.4 Population Proportion 7.5 Inferences About the Difference Between Two Population Means 7.6 Inferences About the Difference Between Two Population Proportions

Hypothesis Testing Hypothesis testing Hypothesis testing for population mean One population mean Two population mean Hypothesis testing for population proportion One population proportion Two population proportions

Hypothesis Testing Hypothesis testing is an inferential statistics method used to determine something about a population, based on the observation of a sample. Information about a population mean in the hypothesis statements will be presented in a form of miu ( ). e.g. “the mean travel time was 40 minutes…”, μ = 40 minutes Assumptions Assumptions needed for validity of Hypothesis Testing The data should be at the interval or ratio level of measurement. Data are a random sample from the population of interest (So that the sample can tell you about the population) The sample mean, is approximately NORMAL (Either the data are figuratively normal or large enough sample size, n, distribution not too skewed)

Hypothesis Testing Hypothesis and test procedure (in general) A statistical test of hypothesis consist of : The Null hypothesis, The Alternative hypothesis, P-value (SPSS) @ Test statistics (manual), α value (SPSS) @ Critical value (manual), The rejection region, The conclusion

Hypothesis Testing Hypothesis and Test Procedures (SPSS) State the null hypothesis, and alternative hypothesis, Compare a P-value (generated by SPSS) with the given value of Make an initial decision (whether to reject or not to reject ) Make the statistical decision and state the managerial conclusion Hypothesis and Test Procedures (manual calculation) Calculate the value of the test statistic Find critical value (using Table Standard Normal III) Compare steps 2 and 3 Make an initial decision (whether to reject or not to reject ) Differences

7.1 Developing Null and Alternative Hypotheses It is not always obvious how the null and alternative hypothesis should be formulated. When formulating the null and alternative hypothesis, the nature or purpose of the test must also be taken into account. We will examine: The claim or assertion leading to the test. The null hypothesis to be evaluated. The alternative hypothesis. Whether the test will be two-tail or one-tail. A visual representation of the test itself. In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier.

7.1 Developing Null and Alternative Hypotheses Null Hypothesis as an Assumption to be Challenged We might begin with a belief on assumption that a statement about the value of a population parameter is true. We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. In these situations, it is helpful to develop the null hypothesis first. When trying to identify the population parameter needed for your solution, look for phrases such as: “It is known that…” “Previous research shows…” “The company claims that…” “A survey showed that…”

7.1 Developing Null and Alternative Hypotheses When writing the Null Hypothesis, make sure it includes an equality symbol. It may look like one of the following: e.g. Example: The label on a soft drink bottle states that it contains at least 500 millilitres. Null Hypothesis: (The label is correct) Alternative Hypothesis: (The label is incorrect) Average tire life is 56000 kilometres. Null Hypothesis: Alternative Hypothesis:

7.1 Developing Null and Alternative Hypotheses Alternative Hypothesis as a Research Hypothesis Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected. When trying to identify the information needed for your Alternative Hypothesis statement, look for the phrases such as: “Is it reasonable to conclude…” “Is there enough evidence to substantiate…” “Does the evidence suggest…” “Has there been a significant…”

7.1 Developing Null and Alternative Hypotheses There are three possible symbols to use in the Alternate Hypotheses, depending on the wording of the question Use “≠” when the question uses words/phrases such as: “is there a difference...?” “is there a change...?” Use “<” when the question uses words/phrases such as: “is there a decrease…?” “is there less…?” “are there fewer…?” Use “>” when the question uses words/phrases such as: “is there a increase…?” “is there more…?” When writing the Alternative Hypothesis, make sure it never includes an “=” symbol. It should look similar to one of the following:

7.1 Developing Null and Alternative Hypotheses Determining null and alternative hypotheses (rule of thumb). Refer previous examples Two-tailed test = Left-tailed test < Right-tailed test >

7.1 Developing Null and Alternative Hypotheses and only will be either rejected or fail to be rejected. Rejecting means that is correct and vice versa. This can be decided by comparing Z and (manual calculation) or by comparing P-value and α (SPSS calculation).

7.2 Type I & Type II Error Type I – Rejecting when it is correct . Type II – Accepting when it is incorrect .

7.3 Population mean (One sample t-test using SPSS) Example (ExamplePM1S.sav) A major oil company developed a petrol additive that was supposed to increase engine efficiency. 22 cars were test driven both with and without the additive and the number of kilometres per litre was recorded. Whether the car was automatic or manual was also recorded and coded as 1= manual and 2 = automatic. During an earlier trial 22 cars were test driven without using the additive. The mean number of kilometres per litre was 10.5. Are the cars that were test driven with additive running more than those without additive? Test the appropriate hypothesis using α=0.05. Probability of committing Type I error

7.3 Population mean (One sample t-test using SPSS) Analysis in SPSS 2 3 Move “withadd” into Test Variables(s) box. Change the to 10.5. Next click “ ” CI % = (1-α)*100. Continue and ok to get an output 1 Output Important values to draw conclusion

7.3 Population mean (One sample t-test using SPSS) Example (ExamplePM1S.sav – solution) 1. State the null hypothesis, and alternative hypothesis, 2. Compare a P-value (generated by SPSS) with the given value of α One- tail: Reject H0 if < α Two- tail: Reject H0 if p-value < α

7.3 Population mean (One sample t-test using SPSS) Example (ExamplePM1S.sav – solution) 3. Make an initial decision (whether to reject or not to reject ) Reject 4. Make the statistical decision and state the managerial conclusion. Hence, it can be concluded that the cars with additive running more than 10.5km/litre, which is more efficient than those without additive.

7.3 Population mean (One sample t-test using SPSS) Exercise (ExercisePM1S.sav) You have been asked to determine whether hypnosis improves memory. 40 men and women are given five minutes to attempt to memorise a list of unrelated words. They are then asked to recall as many as possible while under hypnosis. You performed a study last year with another sample, so you have access to descriptive statistics from a similar group of adults. The mean words recalled in the earlier study, without hypnosis, was 34.6. Determine whether the participants in the present study are comparable with those in the earlier study in terms of recall in normal state. Test at α=0.05.

7.3 Population mean (One sample t-test using SPSS) Extra Exercise The average monthly salary for women in managerial and professional positions is RM2400. Do men in the same positions have average monthly salary that are higher than those for women? A random sample of men in managerial and professional position were interviewed and the SPSS result are as followed: Test the appropriate hypothesis using 0.01 significance level.

7.6 Population mean (two sample t-test using SPSS) In the previous slides, a one-sample t-test was used to determine whether a single sample of scores was likely to have been drawn from a hypothesized population. Next, we are going to test whether two sets of scores using two-sample t-test. There are two kind of two-sample t-test available in SPSS known as the repeated measure t-test and the independent groups t-test. Repeated measure t-test (also known as dependent or paired sample t-test) is used when you have data of two different conditions from only one group of participants (within-subject). It has one additional assumption: The difference between scores for each participant should be normal (applied for small sample size) Independent groups t-test is used when there are data of two different groups that have performed in each of the different conditions. It has two additional assumption: Independence of groups – participants should appear in only one group and the groups should be unrelated. Homogeneity of variance – being determined through another hypothesis testing (hypothesis testing is conducted in the middle of Independent groups t-test analysis.

7.6 Population mean (independent sample t-test using SPSS) Example (ExamplePM1S.sav) On our previous analysis, it has been proven that cars that use fuel additive will experience engine efficiency. But another question arise. Is there any differences in the engine efficiency level of manual and automatic cars that use fuel additive? Test the appropriate hypothesis using α=0.05.

7.6 Population mean (independent sample t-test using SPSS) Analysis in SPSS 2 4 Move “withadd” into Test Variables(s) box and “cartype” into Grouping Variable. Next, click “Define Groups 1 (In Option) CI % = (1-α)*100. Continue and ok to get an output 3 Input “1” in Group 1 and “2” in Group 2. Click Continue

7.6 Population mean (independent sample t-test using SPSS) Analysis in SPSS (Cont.) 5 Important values to draw conclusion

7.6 Population mean (independent sample t-test using SPSS) Example (ExamplePM1S.sav – solution) 1. State the null hypothesis, and alternative hypothesis, 2. Compare a P-value (generated by SPSS) with the given value of α. However, variance equality need to be checked first through Levene’s test ( 2nd assumption - Homogeneity of variance).

7.6 Population mean (independent sample t-test using SPSS) Example (ExamplePM1S.sav – solution) 3. Make an initial decision (whether to reject or not to reject ) Fail to reject 4. Make the statistical decision and state the managerial conclusion. Hence, it can be concluded that there are no difference in the engine efficiency level of manual and automatic cars that use fuel additive.

7.6 Population mean (independent sample t-test using SPSS) Exercise (ExercisePM1S.sav) You have been asked to determine whether hypnosis improves memory. 40 men and women are given five minutes to attempt to memorise a list of unrelated words. They are then asked to recall as many as possible while under hypnosis. Determined whether men and women recall equal numbers of words when under hypnosis. Test at α=0.05.

7.6 Population mean (independent sample t-test using SPSS) Extra Exercise A university conducted an investigation to determine whether car ownership affects academic achievement based on two random samples of 100 male students and their data were input into SPSS. Male students who owned a car were coded as 1 and others as 2 (did not owned a car). Do the data present sufficient evidence to indicate a difference in the mean achievements between car owners and non owners of cars? Test using α = 0.05