Comparing Means from Two Samples and One-Sample Inference for Proportions June 25, 2008 Stat 111 - Lecture 14 - Two Means 1 1

Stat 111 - Lecture 14 - Two Means Administrative Notes Homework 5 is posted on website Due Wednesday, July 1st June 25, 2008 Stat 111 - Lecture 14 - Two Means 2 2

Stat 111 - Lecture 14 - Two Means Outline Two Sample Z-test (known variance) Two Sample t-test (unknown variance) Matched Pair Test and Examples Tests and Intervals for Proportions (Chapter 8) June 25, 2008 Stat 111 - Lecture 14 - Two Means 3 3

Comparing Two Samples Up to now, we have looked at inference for one sample of continuous data Our next focus in this course is comparing the data from two different samples For now, we will assume that these two different samples are independent of each other and come from two distinct populations Population 1:1 , 1 Population 2: 2 , 2 Sample 1: , s1 Sample 2: , s2 June 25, 2008 Stat 111 - Lecture 14 - Means 4

Blackout Baby Boom Revisited Nine months (Monday, August 8th) after Nov 1965 blackout, NY Times claimed an increased birth rate Already looked at single two-week sample: found no significant difference from usual rate (430 births/day) What if we instead look at difference between weekends and weekdays? Sun Mon Tue Wed Thu Fri Sat 452 470 431 448 467 377 344 449 440 457 471 463 405 453 499 461 442 444 415 356 519 443 418 394 399 451 468 432 Weekdays Weekends June 25, 2008 Stat 111 - Lecture 14 - Means 5

Two-Sample Z test We want to test the null hypothesis that the two populations have different means H0: 1 = 2 or equivalently, 1 - 2 = 0 Two-sided alternative hypothesis: 1 - 2  0 If we assume our population SDs 1 and 2 are known, we can calculate a two-sample Z statistic: We can then calculate a p-value from this Z statistic using the standard normal distribution June 25, 2008 Stat 111 - Lecture 14 - Means 6

Two-Sample Z test for Blackout Data To use Z test, we need to assume that our pop. SDs are known: 1 = s1 = 21.7 and 2 = s2 = 24.5 From normal table, P(Z > 7.5) is less than 0.0002, so our p-value = 2  P(Z > 7.5) is less than 0.0004 Conclusion here is a significant difference between birth rates on weekends and weekdays We don’t usually know the population SDs, so we need a method for unknown 1 and 2 June 25, 2008 Stat 111 - Lecture 14 - Two Means 7

Stat 111 - Lecture 14 - Two Means Two-Sample t test We still want to test the null hypothesis that the two populations have equal means (H0: 1 - 2 = 0) If 1 and 2 are unknown, then we need to use the sample SDs s1 and s2 instead, which gives us the two-sample T statistic: The p-value is calculated using the t distribution, but what degrees of freedom do we use? df can be complicated and often is calculated by software Simpler and more conservative: set degrees of freedom equal to the smaller of (n1-1) or (n2-1) June 25, 2008 Stat 111 - Lecture 14 - Two Means 8

Two-Sample t test for Blackout Data To use t test, we need to use our sample standard deviations s1 = 21.7 and s2 = 24.5 We need to look up the tail probabilities using the t distribution Degrees of freedom is the smaller of n1-1 = 22 or n2-1 = 7 June 25, 2008 Stat 111 - Lecture 14 - Two Means 9

Stat 111 - Lecture 14 - Two Means June 25, 2008 Stat 111 - Lecture 14 - Two Means 10

Two-Sample t test for Blackout Data From t-table with df = 7, we see that P(T > 7.5) < 0.0005 If our alternative hypothesis is two-sided, then we know that our p-value < 2  0.0005 = 0.001 We reject the null hypothesis at -level of 0.05 and conclude there is a significant difference between birth rates on weekends and weekdays Same result as Z-test, but we are a little more conservative June 25, 2008 Stat 111 - Lecture 14 - Two Means 11

Two-Sample Confidence Intervals In addition to two sample t-tests, we can also use the t distribution to construct confidence intervals for the mean difference When 1 and 2 are unknown, we can form the following 100·C% confidence interval for the mean difference 1 - 2 : The critical value tk* is calculated from a t distribution with degrees of freedom k k is equal to the smaller of (n1-1) and (n2-1) June 25, 2008 Stat 111 - Lecture 14 - Two Means 12

Confidence Interval for Blackout Data We can calculate a 95% confidence interval for the mean difference between birth rates on weekdays and weekends: We get our critical value tk* = 2.365 is calculated from a t distribution with 7 degrees of freedom, so our 95% confidence interval is: Since zero is not contained in this interval, we know the difference is statistically significant! June 25, 2008 Stat 111 - Lecture 14 - Two Means 13

Stat 111 - Lecture 14 - Two Means Matched Pairs Sometimes the two samples that are being compared are matched pairs (not independent) Example: Sentences for crack versus powder cocaine We could test for the mean difference between X1 = crack sentences and X2 = powder sentences However, we realize that these data are paired: each row of sentences have a matching quantity of cocaine Our t-test for two independent samples ignores this relationship June 25, 2008 Stat 111 - Lecture 14 - Two Means 14

Stat 111 - Lecture 14 - Two Means Matched Pairs Test First, calculate the difference d = X1 - X2 for each pair Then, calculate the mean and SD of the differences d Quantity Sentences Crack X1 Powder X2 Difference d = X1 - X2 5 70.5 12 58.5 25 87.5 18 69.5 100 136 30 106.0 200 169.5 37 132.5 500 211.5 141.0 2000 264 176.5 5000 128.0 50000 52.5 150000 0.0 June 25, 2008 Stat 111 - Lecture 14 - Two Means 15

Stat 111 - Lecture 14 - Two Means Matched Pairs Test Instead of a two-sample test for the difference between X1 and X2, we do a one-sample test on the difference d Null hypothesis: mean difference between the two samples is equal to zero H0 : d= 0 versus Ha : d 0 Usual test statistic when population SD is unknown: p-value calculated from t-distribution with df = 8 P(T > 5.24) < 0.0005 so p-value < 0.001 Difference between crack and powder sentences is statistically significant at -level of 0.05 June 25, 2008 Stat 111 - Lecture 14 - Two Means 16

Matched Pairs Confidence Interval We can also construct a confidence interval for the mean differenced of matched pairs We can just use the confidence intervals we learned for the one-sample, unknown  case Example: 95% confidence interval for mean difference between crack and powder sentences: June 25, 2008 Stat 111 - Lecture 14 - Two Means 17

Summary of Two-Sample Tests Two independent samples with known 1 and 2 We use two-sample Z-test with p-values calculated using the standard normal distribution Two independent samples with unknown 1 and 2 We use two-sample t-test with p-values calculated using the t distribution with degrees of freedom equal to the smaller of n1-1 and n2-1 Also can make confidence intervals using t distribution Two samples that are matched pairs We first calculate the differences for each pair, and then use our usual one-sample t-test on these differences June 25, 2008 Stat 111 - Lecture 14 - Two Means 18

One-Sample Inference for Proportions June 25, 2008 Stat 111 - Lecture 14 - Two Means 19

Stat 111 - Lecture 14- One-Sample Proportions Revisiting Count Data Chapter 6 and 7 covered inference for the population mean of continuous data We now return to count data: Example: Opinion Polls Xi = 1 if you support Obama, Xi = 0 if not We call p the population proportion for Xi = 1 What is the proportion of people who support the war? What is the proportion of Red Sox fans at Penn? June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 20

Inference for population proportion p We will use sample proportion as our best estimate of the unknown population proportion p where Y = sample count Tool 1: use our sample statistic as the center of an entire confidence interval of likely values for our population parameter Confidence Interval : Estimate ± Margin of Error Tool 2: Use the data to for a specific hypothesis test Formulate your null and alternative hypotheses Calculate the test statistic Find the p-value for the test statistic June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 21

Distribution of Sample Proportion In Chapter 5, we learned that the sample proportion technically has a binomial distribution However, we also learned that if the sample size is large, the sample proportion approximately follows a Normal distribution with mean and standard deviation: We will essentially use this approximation throughout chapter 8, so we can make probability calculations using the standard normal table June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 22

Confidence Interval for a Proportion We could use our sample proportion as the center of a confidence interval of likely values for the population parameter p: The width of the interval is a multiple of the standard deviation of the sample proportion The multiple Z* is calculated from a normal distribution and depends on the confidence level June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 23

Confidence Interval for a Proportion One Problem: this margin of error involves the population proportion p, which we don’t actually know! Solution: substitute in the sample proportion for the population proportion p, which gives us the interval: June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 24

Example: Red Sox fans at Penn What proportion of Penn students are Red Sox fans? Use Stat 111 class survey as sample Y = 25 out of n = 192 students are Red Sox fans so 95% confidence interval for the population proportion: Proportion of Red Sox fans at Penn is probably between 8% and 18% June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 25

Hypothesis Test for a Proportion Suppose that we are now interested in using our count data to test a hypothesized population proportion p0 Example: an older study says that the proportion of Red Sox fans at Penn is 0.10. Does our sample show a significantly different proportion? First Step: Null and alternative hypotheses H0: p = 0.10 vs. Ha: p 0.10 Second Step: Test Statistic June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 26

Hypothesis Test for a Proportion Problem: test statistic involves population proportion p For confidence intervals, we plugged in sample proportion but for test statistics, we plug in the hypothesized proportion p0 : Example: test statistic for Red Sox example June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 27

Hypothesis Test for a Proportion Third step: need to calculate a p-value for our test statistic using the standard normal distribution Red Sox Example: Test statistic Z = 1.39 What is the probability of getting a test statistic as extreme or more extreme than Z = 1.39? ie. P(Z > 1.39) = ? Two-sided alternative, so p-value = 2P(Z>1.39) = 0.16 We don’t reject H0 at a =0.05 level, and conclude that Red Sox proportion is not significantly different from p0=0.10 prob = 0.082 Z = 1.39 June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 28

Stat 111 - Lecture 14- One-Sample Proportions Another Example Mass ESP experiment in 1977 Sunday Mirror (UK) Psychic hired to send readers a mental message about a particular color (out of 5 choices). Readers then mailed back the color that they “received” from psychic Newspaper declared the experiment a success because, out of 2355 responses, they received 521 correct ones ( ) Is the proportion of correct answers statistically different than we would expect by chance (p0 = 0.2) ? H0: p= 0.2 vs. Ha: p 0.2 June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 29

Stat 111 - Lecture 14- One-Sample Proportions Mass ESP Example Calculate a p-value using the standard normal distribution Two-sided alternative, so p-value = 2P(Z>2.43) = 0.015 We reject H0 at a =0.05 level, and conclude that the survey proportion is significantly different from p0=0.20 We could also calculate a 95% confidence interval for p: prob = 0.0075 Z = 2.43 Interval doesn’t contain 0.20 June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 30

Stat 111 - Lecture 14- One-Sample Proportions Margin of Error Confidence intervals for proportion p is centered at the sample proportion and has a margin of error: Before the study begins, we can calculate the sample size needed for a desired margin of error Problem: don’t know sample prop. before study begins! Solution: use which gives us the maximum m So, if we want a margin of error less than m, we need June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 31

Margin of Error Examples Red Sox Example: how many students should I poll in order to have a margin of error less than 5% in a 95% confidence interval? We would need a sample size of 385 students ESP example: how many responses must newspaper receive to have a margin of error less than 1% in a 95% confidence interval? June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 32

Stat 111 - Lecture 14- One-Sample Proportions Next Class - Lecture 15 Two-Sample Inference for Proportions Moore, McCabe and Craig: Section 8.2 June 25, 2008 Stat 111 - Lecture 14- One-Sample Proportions 33 33