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3. Yellow Card on the Big 5 Parameters We have covered most of the top 3 columns page 133

4. The Paired Data Scenario (page 135) Ways that paired data can occur: Each person/unit is measured twice. Two measurements of same characteristic made under different conditions. Example: measuring a quantitative response both before and after treatment. Similar individuals/units paired prior to experiment. Each member of a pair receives a different treatment. Same (quantitative) response variable is measured for all individuals.

5. Paired Design Focus = On the Differences Consider one popul of all possible differences  Parameter:  d = mean of popul of differences We have a sample from this population  Data: d 1, d 2, …, d n Think about it: What will we compute from data?

6. Example of Paired Data Scenario (text page 337) From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

7. 9.7 SD Module 4: Sampling Distribution for the Sample Mean of Paired Differences Identify: Population = Parameter = Sample = Statistic = Freshmen Weight (Example 9.13): Study to learn about average weight gain in the first year of college for students. Sample of 60 students resulted in an average weight gain of 4.2 pounds (over the first 12 weeks of college).

8. Sampling Distribution for a Sample Mean Difference Can say how close this obs sample mean difference of 4.2 pounds is to the true population mean difference  d ? _____ If take another random sample of same size, would we get same value for sample mean difference? ______________ What are possible values for if we took many random samples of the same size from this population? What would the distribution of the possible values look like? What can we say about the distribution of the sample mean difference? Don’t we already know?

9. Distribution of the Sample Mean Difference Let  d = mean for population of interest. Let  d = standard deviation for popul of interest. Let = sample mean for random sample of size n. If the population of differences has a normal distribution (bell-shaped), and a random sample of any size is obtained, then the distribution of the sample mean difference, is also normal …

10. Distribution of the Sample Mean Difference Let  d = mean for population of interest. Let  d = standard deviation for popul of interest. Let = sample mean for random sample of size n. If the population of differences is NOT normal distribution (bell-shaped), but a large random sample of size n is obtained, then the distribution of the sample mean difference, is approximately normal … This is a variation of the Central Limit Theorem (CLT).

11. Notes about Distribution of Sample Mean Difference page 137 (1) Arbitrary level for ‘large’ = 30. However, if any differences are extreme outliers, better = larger. (2) Standard deviation of is measure of accuracy of the process of using sample mean difference to estimate population mean difference. (3) In practice, popul standard deviation  d rarely known, so sample standard deviation s d used  result is a standard error.

12. Big Idea about CI for Population Mean Difference Use the sample mean difference and its standard error to produce a range of reasonable values for the population mean difference …  (a few)s.e.( ) Looking ahead: Do you think the ‘few’ will be a z* value or a t* value? What degrees of freedom will be used?

13. Big Idea about Testing about Popul Mean Difference Use the sample mean difference and its standard error to produce a standardized test statistic for testing hypotheses about the population mean difference … Sample statistic – Null value (Null) standard error Looking ahead: Do you think the test statistic will be a z statistic or a t statistic? Most common null value used in the null hypothesis? H 0 :  d = _______

14. 11.3 CI for Popul Mean of Paired Differences (page 139) Population Parameter:  d Data: d 1, d 2, …, d n Sample Estimate: Standard Error: Sample Estimate  Multiplier x Std error

15. CI for the Population Mean Difference page 139 One-sample t Confidence Interval for  d where t* is from a t(n – 1) distribution. Interval requires that the differences can be considered a random sample from normal population. If sample size large, assumption of normality not so crucial and result is approximate.

16. Try It! Changes in Reasoning Scores page 140 Do piano lessons improve spatial-temporal reasoning of preschool children? Data: Change in reasoning score, after lessons – before lessons, with larger values = better reasoning, for a random sample of n = 34 preschool children. 257-227410734 34945296036 3467-27-3344 What do you see?Think about it … what should we do with data first?

17. Learning about changes in reasoning scores … (a) Display data, summarize the distribution. Notes: 1.Diff = after – before so … 2.Sample mean difference = _______ 3.Normality of the response for the population? Change in Reasoning

18. Learning about changes in reasoning scores … Some summary measures.

19. Learning about changes in reasoning scores (b)Give a 95% confidence interval for the mean improvement in reasoning scores.

20. Learning about changes in reasoning scores 95% CI for popul mean difference in reasoning score: (2.56, 4.68) (c)What value is of particular interest to see whether or not it is in the interval? Get Ready to click in for part (d)

21. Click in: Yes or No? (d)A student in your class wrote the following interpretation about 95% confidence level used in making the interval. Is it a correct interpretation? “If this study were repeated many times, we would expect 95% of the resulting confidence intervals to contain the sample mean improvement in reasoning scores.”

22. SPSS Note: Differences already computed and entered. With SPSS we perform a one-sample t-test on differences and specify test value of 0. Options box confidence level = 95% If the before and after scores were entered into SPSS, then we would perform a paired t-test option. SPSS would compute the differences for us and provide the confidence interval results. Details of the SPSS steps for analyzing paired data = PreLab 6 and SPSS Module 6 of your Lab Workbook.

23. Inference Methods (yellow card) or page 145

24. Recall: Changes in Reasoning Scores page 140 Do piano lessons improve spatial-temporal reasoning of preschool children? Data: Change in reasoning score, after– before, for a random sample of n = 34 preschool children. 95% CI for popul mean difference in reasoning score: (2.56, 4.68)

25. SPSS Note: Differences already computed and entered. With SPSS we perform a one-sample t-test on differences and specify test value of 0. Options box confidence level = 95% If the before and after scores were entered into SPSS, then we would perform a paired t-test option. SPSS would compute the differences for us and provide the confidence interval results. Details of the SPSS steps for analyzing paired data = PreLab 6 and SPSS Module 6 of your Lab Workbook.

26. Paired Design Focus = On the Differences Consider one popul of all possible differences  Parameter:  d = mean of popul of differences We have a sample from this population  Data: d 1, d 2, …, d n Let’s turn to hypothesis testing about the population mean difference  d

27. 13.3 Testing about a Population Mean Difference  d Page 143: Just a one-sample t test on the differences 1.H 0 :H a : 2.H 0 : H a : 3.H 0 : H a :

28. Test Statistic and Conditions Test statistic = Sample statistic – Null value Standard error If H 0 is true, this test statistic has a __________ distrib. Use this distribution to find the p-value. Conditions: The observed differences are a random sample from a normally distributed population.

29. Try It! Knob Turning page 144 Background: n = 25 right-handed students and device with two different knobs (right-hand thread and left-hand thread). Response: time to move knob fixed distance. Are RH threads easier to turn on average? (Use a 5% significance level.) a. Why is this a paired design and how should randomization be used in the experiment? Think about part (b) stating the hypotheses– be ready to clicker in answer.

30. Are right threads easier to turn on average? (b) State the hypotheses. A) H 0 :  d = 0 vs H a :  d ≠ 0 B) H 0 :  d = 0 vs H a :  d < 0 C) H 0 :  d = 0 vs H a :  d > 0

31. Try It! Knob Turning c. Perform the test.H 0 :  d = 0 vs H a :  d < 0

32. Try It! Knob Turning d. Which of the following are an assumption for paired t-test? the turning times for the right-hand threaded knob are independent of the turning times for the left-hand threaded knob. the turning times for right-hand threaded knob are normally distributed. the difference in turning times (diff = RT – LT) is normally distributed.