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Testing Hypotheses about the Difference Between the Means of Two Populations STAT Chapter 9 METH Chapter 13 S07 Review: Hypothesis testing steps 1. Choose the appropriate test (statistical model) 2. Create the Null and Alternative Hypotheses 3. State the criteria for rejecting Null 4. Do the calculations 5. Make the decision: Reject Null or Not 6. Draw Conclusions 7. Write the results up in an APA-style format (Steps 1, 2, & 3 constitute “Setting up the Hypothesis Test”) Example: Recently, an endogenous brain neurotransmitter (NT) called Galanin has been discovered that appears to specifically affect one’s desire to eat foods with high fat content. The more of this naturally occurring NT that an individual possesses, the higher is his or her craving for high-fat foods. A drug company developed a drug that blocks Galanin without affecting the appetite for healthier (less fat) foods. They hope it will be useful for obesity. After approval by the FDA, one of the neuroscientists of the drug company conducted an experiment in which 15 obese female volunteers were randomly selected and given the experimental drug for 6 months. Baseline and ending (6 month) weights were recorded for each subject. 1 Subject Baseline Ending The data for Number Weight Weight 1 165 145 the example: 2 143 137 3 175 170 4 135 136 5 148 141 6 155 138 7 158 137 8 140 125 9 172 161 10 164 156 11 178 165 12 182 170 13 190 176 14 169 154 15 157 143 REVIEW: Hypothesis testing Step 1 Choose the sampling distribution or statistical model First, ask what type of data: nominal or interval/ratio? Second, ask what question are you interested in answering: relationship or difference? Our answers (at the moment) are interval/ratio data and differences: therefore, z, t, or F tests (see Flowchart for choosing correct statistics) Review: Step 1 continued Next question is: How many conditions (or samples or groups)? If you have 1 sample of data, then you must choose between a z-test or a one-sample t-test If there are 2 conditions (or 2 samples of data), then you must choose between the two types of two-sample t-tests If there are 3 conditions/samples (or more), then you must choose the appropriate F-test or ANOVA (we’ll cover these tests later ☺) 2 One-sample hypothesis tests compare one sample mean to a population mean, in order to determine the likelihood that the sample is part of that population, or whether it has been drawn from a different population (are they statistically different?) z-tests: t-tests: Must know μ (mean) Must know μ (mean) of and σ (standard the population data deviation) of the Use s (standard population data deviation of the sample data) to estimate the standard error of the mean Two-sample t-tests are used to determine whether the means of two groups of scores differ from each other to a statistically significant degree Whenever there is more than 1 condition, or sample of data, the next question to ask is what type of subject design was used? Independent groups vs. Repeated measures There are 2 types of two-sample t-tests One is used for One is used for these these designs: designs: Random groups Repeated measures designs designs Natural groups Matched groups designs designs Multiple names: Multiple names Repeated measures Independent Related samples groups Correlated samples Independent Dependent samples samples Paired samples Matched samples 3 Hypothesis testing: Step 1 What statistical test do we use? Two- t- Two-sample t-test Which one? Repeated measures Review: Hypothesis testing steps 1. Choose the appropriate test (statistical model) 2. Create the Null and Alternative Hypotheses 3. State the criteria for rejecting Null 4. Do the calculations 5. Make the decision: Reject Null or Not 6. Draw Conclusions 7. Write the results up in an APA-style format (Steps 1, 2, & 3 constitute “Setting up the Hypothesis Test”) Step 2: Null and alternative hypotheses? Null H0: µ1 = µ2 H0: µD = 0 Alternative H1: µ1 ≠ µ2 H1: µD ≠ 0 4 Step 3: Criteria for rejecting Null? Use standard choices for Psychologists: α = .05 Two-tailed test Determine critical value of t: df = N – 1, where N is the number of pairs df = 15 – 1 = 14 Check Appendix in your textbook for the critical value of t corresponding to the above info: tcritical = 2.145 Step 4: Do the calculations Enter the data into SPSS CRITICAL POINT: If more than 1 data point is obtained from each subject (or when you’ve matched subjects), all the data goes on the same row for a given subject and each column is labeled If there are 2 samples of data, but each subject only gives 1 piece of data, then you have 1 column for the data itself, and a second column to indicate which condition the subject is in Subject Baseline Ending The data for Number Weight Weight 1 165 145 the example: 2 143 137 3 175 170 4 135 136 5 148 141 6 155 138 7 158 137 8 140 125 9 172 161 10 164 156 11 178 165 12 182 170 13 190 176 14 169 154 15 157 143 5 Step 4: Do the calculations, cont’d. Double-check to make sure that the data was entered correctly Click on Analyze Then Compare Means, Then Paired-Samples t-test, Click on each variable to be analyzed in this test, then click on the arrow in the middle of the two boxes Click on OK SPSS output Paired Samples Statistics Std. Error Mean N Std. Deviation Mean Pair baseline 162.0667 15 16.10442 4.15814 1 endweight 150.2667 15 15.42478 3.98266 Paired Samples Test Paired Differences 95% Confidence Interval of the Std. Error Difference Mean Std. Deviation Mean Lower Upper t df Sig. (2-tailed Pair 1 baseline - endwe 1.80000 5.93055 1.53126 8.51577 5.08423 7.706 14 .000 Step 5: Make the decision Compare the tobtained (just calculated) to the tcritical in order to make the decision: tobtained = 7.706 and tcritical = 2.145 Is tobtained larger than tcritical ? Yes. Reject the null. 6 Step 6: Draw conclusions There is a significant difference between baseline weight and ending weight. Do I need more info ? Yes. The average baseline weight was 162.07 and the average ending weight was 150.27. Therefore, the women on the drug lost an average of 11.8 lbs. You’ll need a graph: Click on Graphs Choose Bar Choose Simple, and at the bottom, click on Summaries of separate variables, then click on Define Click on both variables on the left, then click on the arrow, then choose OK Go into Chart Editor (by double-clicking somewhere on the Graph itself), and re- format the graph in APA style X out of Chart editor, then copy and paste the graph into a word document Step 7: Write it up in APA-style A paired-samples t-test was used to compare the participants’ weight before beginning (baseline) and again after 6 months on the drug that was purported to block cravings for high-fat foods. The participants lost a significant amount of weight (11.8 lbs) on the drug across the six months, t(14) = 7.71, p < .001. The data are graphed in Figure 1. 7 Example 2 A psychologists is interested in determining whether immediate memory capacity is affected by sleep loss. Immediate memory is defined as the amount of material that can be remembered immediately after it has been presented. Twelve students are randomly selected from Introductory Psychology and randomly assigned to 2 groups. One of the groups is sleep- deprived for 24 hours before the material is presented. All subjects in the other group receive the normal amount of sleep (7-8 hours). The material consists of a series of slides, with each containing nine numbers. Each slide is presented for a short time interval (50 ms) after which the subject must recall as many numbers as possible. The scores represent the percentage correctly recalled. Review: Hypothesis testing steps 1. Choose the appropriate test (statistical model) 2. Create the Null and Alternative Hypotheses 3. State the criteria for rejecting Null 4. Do the calculations 5. Make the decision: Reject Null or Not 6. Draw Conclusions 7. Write the results up in an APA-style format (Steps 1, 2, & 3 constitute “Setting up the Hypothesis Test”) Hypothesis testing: Step 1 What statistical test do we use? Two- t- Two-sample t-test Which one? Independent samples 8 Step 2: Null and alternative hypotheses? Null H0: µ1 = µ2 or X1 = X2 Alternative H1: µ1 ≠ µ2 or X1 ≠ X2 Step 3: Criteria for rejecting Null? Use standard choices for Psychologists: α = .05 Two-tailed test Determine critical value of t: df = N1 + N2 – 2 df = 12 – 2 = 10 Check Appendix in your textbook for the critical value of t corresponding to the above info: tcritical = 2.228 Step 4: Do the calculations Enter the data into SPSS CRITICAL POINT: If more than 1 data point is obtained from each subject (or when you’ve matched subjects), all the data goes on the same row for a given subject and each column is labeled If there are 2 samples of data, but each subject only gives 1 piece of data, then you have 1 column for the data itself, and a second column to indicate which condition the subject is in 9 Data for Subject Group Percent Example 2: Number Recalled 1 1 68 2 1 73 3 1 72 Group 1 = 4 1 65 non-deprived 5 1 70 control group 6 1 73 7 2 70 8 2 64 Group 2 = 9 2 68 sleep-deprived 10 2 63 11 2 69 group 12 2 66 Step 4: Do the calculations, cont’d. Double-check to make sure that the data was entered correctly Choose Analyze Click on Compare Means Choose Independent-Samples t-test, Put column with data in Test Variable box and column with group in Grouping Variable box Click on Define Groups, enter correct numbers Then click on OK SPSS output Group Statistics Std. Error 1=control,2=no sleep N Mean Std. Deviation Mean percent 1.00 6 70.1667 3.18852 1.30171 2.00 6 66.6667 2.80476 1.14504 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Mean Std. Error Difference F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper percent Equal variances .042 .842 2.019 10 .071 3.50000 1.73365 -.36282 7.36282 assumed Equal variances 2.019 9.840 .072 3.50000 1.73365 -.37136 7.37136 not assumed 10 Step 5: Make the decision Compare the tobtained (just calculated) to the tcritical in order to make the decision: Is tobtained larger than tcritical ? tobtained = 2.019 vs. tcritical = 2.228 No. Do not reject the null. Step 6: Draw conclusions There is not a significant difference in percent of numbers remembered between subjects that were sleep deprived versus those that were not. Do I need more info ? Yes. Non-sleep-deprived subjects remembered an average of 70.17%, whereas sleep-deprived subjects remembered an average of 66.67% of the numbers. You’ll want a graph of the data… Click on Graphs Choose Simple, then click on Summaries for groups of cases (at bottom of window), then Define Click on Other Statistic (e.g., mean) under Bars Represent box Then choose your data column on the left, and click on the arrow in the Bars Represent box to move it in as the Variable Click on the group column on the left, then click on the arrow to move it into the Category Axis box Click on OK Go into Chart Editor and format it for APA style Copy and paste it into your word document 11 Step 7: Write it up in APA-style The percentage of numbers recalled correctly for sleep- deprived versus non-sleep- deprived students was analyzed with an independent-samples t- test. The percent recalled for the sleep-deprived participants did not differ significantly (t(10) = 2.02, p = .071) from the percent recalled by the participants that slept the normal amount. As can be seen in the graph (see Figure 1), the two groups did not differ. Underlying Assumptions for t-tests In theory: The data should be normally distributed within each population The variances of the two populations should be equal In practice: You can generally ignore the assumption of normally distributed data in the population You can ignore the assumption about equal variances as well, as long as the number of subjects in each condition is approximately equal Confidence Limits on MD Paired-samples t-test: XD ± (tcritical) ( sD) Where: MD = mean of the difference scores sD = standard error of the mean of the differences 12 Confidence Limits on µ1 - µ2 Independent-samples t-test (X1 – X2) ± (tcritical) ( s x1-x2) Standard error of the difference between means = (N1 – 1) s12 + (N2 – 1) s22 (1/N1 + 1/N2) √ N1 + N2 - 2 Confidence limits for the 2 examples Example 1: Example 2: 11.8 ± (2.145) (1.53126) (70.1667 – 66.6667) = ± (2.228) (1.733652) 11.8 ± 3.2846 = = 3.5 ± 3.86258 8.5154 – 15.0846 = -.3626 – 7.3626 13

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null hypothesis, test statistic, hypothesis testing, standard error, confidence interval, sampling distribution, the difference, hypothesis test, degrees of freedom, standard deviation, alternative hypothesis, population mean, population variances, critical value, sample size

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posted: | 5/8/2010 |

language: | English |

pages: | 13 |

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