Lecture 37 Sections 11.1, 11.2, Wed, Nov 4, Hampden-Sydney College. Paired Samples. Robb T. Koether. Homework Review.

Transcription

1 Lecture 37 Sections 11.1, 11.2, 11.3 Hampden-Sydney College Wed, Nov 4, 2009

2 Outline

3 Outline

4 Exercise 10.21, page 648. The value reported as lost for a random sample of n = 20 pickpocket offenses occurring in a city is given here

5 Exercise 10.21, page 648. (a) Use the data to construct a 95% confidence interval for the mean value lost in all pickpocket offenses for this city. (b) What is the margin of error for the interval estimate in part (a)? (c) Give an interpretation of the interval and of the confidence level.

6 Solution (a) Use the TI-83 to find the mean and standard deviation of the sample. x = s = Use the t table to find t 19,0.025 = The confidence interval is ( ) ( ) s x ± t = ± n 20 = ±

7 Solution (b) The margin of error is (c) The interpretation is that if we repeated this procedure many times, about 95% of our confidence intervals would contain the true value of µ.

8 Outline

9 In this chapter we will consider problems that compare two populations.

10 In this chapter we will consider problems that compare two populations. For example, we could compare

11 In this chapter we will consider problems that compare two populations. For example, we could compare The proportion of men who vote Republican the proportion of women who vote Republican. (p 1 p 2.)

12 In this chapter we will consider problems that compare two populations. For example, we could compare The proportion of men who vote Republican the proportion of women who vote Republican. (p 1 p 2.) The average age at which a male first marries the average age at which a female first marries. (µ 1 µ 2.)

13 In this chapter we will consider problems that compare two populations. For example, we could compare The proportion of men who vote Republican the proportion of women who vote Republican. (p 1 p 2.) The average age at which a male first marries the average age at which a female first marries. (µ 1 µ 2.) The bad news: Things will get a bit more complicated.

14 In this chapter we will consider problems that compare two populations. For example, we could compare The proportion of men who vote Republican the proportion of women who vote Republican. (p 1 p 2.) The average age at which a male first marries the average age at which a female first marries. (µ 1 µ 2.) The bad news: Things will get a bit more complicated. The good news: You will not need to memorize any more formulas.

15 Outline

16 Definition (Bivariate data, paired data) Bivariate data are data in which each datum is a pair of observations. These are also called paired data. Typically the two values are called x and y. Definition ( samples, dependent samples) If the data are paired, then the sample of x values and the sample of y values are called paired samples or dependent samples.

17 data are often before and after observations. By comparing the mean before treatment to the mean after treatment, we can determine whether the treatment had an effect. To make direct comparisons of the two samples, they must be measuring the same sort of thing. Clearly, paired samples must be of the same size.

18 Example (High-School Graduation Rates) Graduation Rate 2008 Graduation Rates for Richmond-area High Schools High School

19 Example (High-School Graduation Rates) Graduation Rate 2009 Graduation Rates for Richmond-area High Schools High School

20 Example (High-School Graduation Rates) Graduation Rate Graduation Rates for Richmond-area High Schools High School

21 Example (High-School Graduation Rates) Graduation Rate Graduation Rates for Richmond-area High Schools High School

22 Example (High-School Graduation Rates) Change in Graduation Rates for Richmond-area High Schools Change ingraduation Rate High School -15

23 Example (High-School Graduation Rates) Was there an overall improvement in the graduation rate? That is, is the average difference greater than 0? See the article Va. graduation, dropout rates improve over last year.

24 On the other hand, with independent samples, we simply take one sample from one population and another sample from another population. There is no logical way to pair the data. Furthermore, the independent samples could be of different sizes. In this chapter, we will first study paired data. Then we will study independent samples.

25 Outline

26 Let the pairs be denoted (x 1, x 2 ). Let d = x 2 x 1. We will study the case where d has a normal distribution. Let µ D denote the mean of this distribution and σ D denote the standard deviation.

27 Hypothesis Tests Concerning µ D The only null hypothesis for µ D that we will consider is H 0 : µ D = 0. We will consider any of the three alternatives H 1 : µ D < 0. H 1 : µ D > 0. H 1 : µ D 0.

28 Hypothesis Tests Concerning µ D For large samples, the test statistic is z = d 0 s D / n. For small samples it is necessary that d have a normal distribution. Then the test statistic is t = d 0 s D / n.

29 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) Suppose that a group of 10 students take a math placement test. Let the variable x 1 represent their scores on that test. Then they are given an Algebra refresher course and they retake the placement test. Let the variable x 2 represent their scores on the retest.

30 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) The following table shows the results Student 1st Score (x 1 ) 2nd Score (x 2 ) Difference (d)

31 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) The following table shows the results Student 1st Score (x 1 ) 2nd Score (x 2 ) Difference (d)

32 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) Test the hypothesis, at the 10% level, that the refresher course improved their grades on the placement test.

33 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) (1) Let x 1 be the first test score, let x 2 be the second test score, and let d = x 2 x 1. Then the hypotheses are H 0 : µ D = 0. H 1 : µ D > 0. (2) α = (3) Let t = d 0 s D / n.

34 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) (4) Compute the value of the test statistic. Enter the x 1 values into L 1 and the x 2 values into L 2. Evaluate the difference L 2 L 1 and store it in L 3. Use 1-Var Stats L 1 to get d and s D. We find that d = 3 and s D = Then 3 t = 5.354/ 10 = =

35 Hypothesis Tests Concerning µ D Example (Hypothesis Tests Concerning µ D ) (5) p-value = tcdf(1.772,e99,9) = (6) Reject H 0. (7) Students scores on the placement test are higher after taking the Algebra refresher course.

36 Graduation Rates Example (Graduation Rates ) Test whether there has been a significant change in the graduation rates of the Richmond-area schools from 2008 to 2009.

37 Outline

38 Read Sections 11.1, 11.2, 11.3, pages Let s Do It! 11.1, 11.2, Exercises 1-8, page 676. Exercises 9-14, page 689.

39 Answers 2. (a) samples. The was no attempt to pair each male with a female. (b) Perhaps males are more likely to drop courses in which they are having difficulty. (That is pure speculation.) 4. Dogs 2, 7, 8, 9, 10 in one group. The rest in the other group.

40 Answers 6. (a) Collect a sample of sophomores and an independent sample of freshmen. Find the average number of times the members of each sample sought the advice of their advisor. (b) Collect a sample of individuals who have not taken the Kaplan SAT prep course administer to them the SAT test to get their scores. Then have them take the Kaplan prep course. Then re-administer the SAT test and see if their scores increased. This would be a paired sample (each person paired with himself, before and after). (c) Collect a sample of males and an independent sample of females. Find the average number of hours per week that each group studies.

41 Answers 8. (a) The placebo effect is the phenomenon of patients improving because they believe they are being given a drug when in fact they are being given a placebo. (b) To eliminate the confounding variable that one group knew it was being given the medication and, without the placebo, the other group would know that it was not being given the medication. (c) samples. There would be no logical way to pair the members. (d) The p-value would be larger than The report says that the medication had no clinical advantage over a placebo. That is, the results were not significant at the 5% level. (e) H 0 : µ 1 = µ 2. H 1 : µ 1 > µ 2.

42 Answers 10. (a) 12. (a) 14. (a)