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3. 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.

4. 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.

5. 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.

6. The Independent Samples Scenario (page 147) Ways that independent samples can occur: Random samples are taken separately from two populations and same response variable is recorded for each individual. One random sample is taken and a variable recorded for each individual, but then units are categorized as belonging to one population or another (e.g. M/F). Participants randomly assigned to one of two treatment conditions, same response variable recorded for each unit. Quantitative Response => look at difference between two means

7. Example of Indep Samples Scenario (from text pg 337) Practice: For each of following studies: Click in … Is it based on paired or independent samples? From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

8. 1. Are average GPAs the same for Psychology majors versus Economics majors, if we take a random sample of 30 Psych majors and a random sample of 30 Econ majors? A) Paired B) Independent

9. 2.On average, is the number of pedestrians crossing the intersection higher in the morning than in the afternoon if we measure the counts for each morning and afternoon time period for 20 randomly sampled days this month? A) Paired B) Independent

10. 3.Is the average car speed lower if a police car is parked on side than the average car speed if the police car patrols the area? Speeds for 25 randomly selected cars will be measured during a Monday morning hour when a police car is parked, and then speeds for 25 randomly selected cars will be measured during the next Monday morning hour when a police car is patrolling. A) Paired B) Independent

11. 9.8 SD Module 5: Sampling Distribution for the Difference Between Two Sample Means (page 154) Typical Summary:  1 = population mean fastest speed for male college student popul.  2 = population mean fastest speed for female college student popul. Want to learn about  1 –  2 … How to estimate  1 –  2 ? With Who are the speed demons? (Example 9.14): Survey of college students : “What is the fastest you have ever driven a car? _____ mph”.

12. Sampling Distribution for Difference in Sample Means Can anyone say how close = 19 mph is to the true difference  1 –  2 ? If we repeat this survey, would we get the same value for ? Is 19 mph convincingly large enough? What are possible values for ?

13. Distribution of Difference of Sample Means Recall from Section 8.8: Mean of difference = difference in means Variance of difference = sum of variances* (*if indep) Recall from Section 9.6:: Standard deviation of = which implies the Variance of =

14. Distribution of Difference of Sample Means If the two populations are normally distributed (or sample sizes both large enough), then is (approximately)…

15. Standard Error for the Difference in Two Sample Means page 149 where s 1 and s 2 are two sample std devs The standard error of estimates, roughly, the average distance of the possible values from  1 –  2.

16. Big Idea: CI for Difference in Population Means Use the difference in sample means and its standard error to produce a range of reasonable values for the difference in population means …  (a few)s.e.( ) THINK ABOUT IT: Do you think the ‘few’ will be a z* value or a t* value? What do you think will be the degrees of freedom?

17. Big Idea: Testing about Difference in Popul Means Use the difference in sample means and its standard error to produce a standardized test statistic for testing hypotheses about the difference in population means … Sample statistic – Null value (Null) standard error THINK ABOUT IT: Do you think the test statistic will be a z statistic or t statistic? Most common null value used? H 0 :  1 –  2 = _______

18. Which of the following is the latest statistic that again has (approx) a normal distribution? A) The difference B) The difference m 1 – m 2

19. 11.4 CI for Difference in Two Popul Means (pg 151) Lesson 1: The General (Unpooled) Case Typical Summary of Responses for a Two Independent Samples Problem:  1 = population mean for first population  2 = population mean for second population Parameter =  1 –  2 Estimate = Standard error = PopulationSample Size Sample Mean Sample Standard Deviation 1 n1n1 2 n2n2

20. CI for the Difference in Population Means General Two Independent-Samples t Confidence Interval for  1 -  2 where and t* is an appropriate value from a t distribution. df using an approx or conservatively as df = smaller of (n 1 – 1, n 2 – 1). This interval requires we have independent random samples from normal populations. If the sample sizes are large (both > 30), the assumption of normality is not so crucial and result is approx.

21. 11.4 CI for Difference in Two Popul Means Lesson 2: The Pooled Case Reasonable to assume measurements in two populations have same variances so that ____________________ where ________ denotes common population variance Then use both sample standard deviations to get one overall estimate of the common population variance.

22. 11.4 CI for Difference in Two Popul Means Lesson 2: The Pooled Case The pooled standard deviation: Notes: 1.Each sample variance weighted by corresponding df. Larger sample = larger weight 2.Denominator = total df df =

23. 11.4 CI for Difference in Two Popul Means Lesson 2: The Pooled Case Unpooled Replacing individual standard deviations with pooled version in the standard error leads to: Pooled s.e.( ) =

24. Pooled CI for Difference in Population Means Pooled Two Independent-Samples t Confidence Interval for  1 -  2 where and and t* is the value from a t (n 1 + n 2 – 2) distribution. This interval requires we have independent random samples from normal populations with equal population variances. If the sample sizes are large (both > 30), the assumption of normality is not so crucial and the result is approximate.