CHAPTER 10 Comparing Two Populations or Groups

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1 CHAPTER 10 Comparing Two Populations or Groups 10.2 Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers

2 Comparing Two Means Learning Objectives After this section, you should be able to: DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample means. DETERMINE whether the conditions are met for doing inference about µ 1 µ 2. CONSTRUCT and INTERPRET a confidence interval to compare two means. PERFORM a significance test to compare two means. DETERMINE when it is appropriate to use two-sample t procedures versus paired t procedures. The Practice of Statistics, 5 th Edition 2

3 Introduction What if we want to compare the mean of some quantitative variable for the individuals in Population 1 and Population 2? Our parameters of interest are the population means µ 1 and µ 2. The best approach is to take separate random samples from each population and to compare the sample means. Suppose we want to compare the average effectiveness of two treatments in a completely randomized experiment. We use the mean response in the two groups to make the comparison. The Practice of Statistics, 5 th Edition 3

4 The Sampling Distribution of a Difference Between Two Means To explore the sampling distribution of the difference between two means, let s start with two Normally distributed populations having known means and standard deviations. Based on information from the U.S. National Health and Nutrition Examination Survey (NHANES), the heights (in inches) of ten-year-old girls follow a Normal distribution N(56.4, 2.7). The heights (in inches) of ten-year-old boys follow a Normal distribution N(55.7, 3.8). Suppose we take independent SRSs of 12 girls and 8 boys of this age and measure their heights. What can we say about the difference x f - x m in the average heights of the sample of girls and the sample of boys? The Practice of Statistics, 5 th Edition 4

The Sampling Distribution of a Difference Between Two Means Using Fathom software, we generated an SRS of 12 girls and a separate SRS of 8 boys and calculated the sample mean heights.

5 The Sampling Distribution of a Difference Between Two Means Using Fathom software, we generated an SRS of 12 girls and a separate SRS of 8 boys and calculated the sample mean heights. The difference in sample means was then be calculated and plotted. We repeated this process 1000 times. The results are below: What do you notice about the shape, center, and spread of the sampling distribution of x f - x m? The Practice of Statistics, 5 th Edition 5

6 The Sampling Distribution of a Difference Between Two Means Both x 1 and x 2 are random variables. The statistic x 1 - x 2 is the difference of these two random variables. In Chapter 6, we learned that for any two independent random variables X and Y, 2 m X -Y = m X - m Y and s X -Y =s 2 2 X +s Y The Sampling Distribution of the Difference Between Sample Means Choose an SRS of size n 1 from Population 1 with mean µ 1 and standard deviation σ 1 and an independent SRS of size n 2 from Population 2 with mean µ 2 and standard deviation σ 2. Shape When the population distributions are Normal, the sampling distribution of x 1 - x 2 is approximately Normal. In other cases, the sampling distribution will be approximately Normal if the sample sizes are large enough (n 1 ³ 30,n 2 ³ 30). Spread The standard deviation of the sampling distribution of x 1 - x 2 is s s 2 n 1 n 2 as long as each sample is no more than 10% of its population (10% condition). The Practice of Statistics, 5 th Edition 6

7 The Sampling Distribution of a Difference Between Two Means The Practice of Statistics, 5 th Edition 7

8 The Two-Sample t Statistic When data come from two random samples or two groups in a randomized experiment, the statistic x 1 - x 2 is our best guess for the value of m 1 -m 2. If the Normal condition is met, we standardize the observed difference to obtain a t statistic that tells us how far the observed difference is from its mean in standard deviation units. The Practice of Statistics, 5 th Edition 8

9 The Two-Sample t Statistic t = (x 1 - x 2 ) - (m 1 - m 2 ) s 1 2 n 1 + s 2 2 n 2 The two-sample t statistic has approximately a t distribution. We can use technology to determine degrees of freedom OR we can use a conservative approach, using the smaller of n 1 1 and n 2 1 for the degrees of freedom. The Practice of Statistics, 5 th Edition 9

10 The Two-Sample t Statistic Conditions for Performing Inference About µ 1 - µ 2 Random: The data come from two independent random samples or from two groups in a randomized experiment. o 10%: When sampling without replacement, check that n 1 (1/10)N 1 and n 2 (1/10)N 2. Normal/Large Sample: Both population distributions (or the true distributions of responses to the two treatments) are Normal or both sample sizes are large (n 1 30 and n 2 30). If either population (treatment) distribution has unknown shape and the corresponding sample size is less than 30, use a graph of the sample data to assess the Normality of the population (treatment) distribution. Do not use two-sample t procedures if the graph shows strong skewness or outliers. The Practice of Statistics, 5 th Edition 10

11 Confidence Intervals for µ 1 µ 2 Two-Sample t Interval for a Difference Between Two Means The Practice of Statistics, 5 th Edition 11

12 Significance Tests for µ 1 µ 2 An observed difference between two sample means can reflect an actual difference in the parameters, or it may just be due to chance variation in random sampling or random assignment. Significance tests help us decide which explanation makes more sense. The null hypothesis has the general form H 0 : µ 1 - µ 2 = hypothesized value We re often interested in situations in which the hypothesized difference is 0. Then the null hypothesis says that there is no difference between the two parameters: H 0 : µ 1 - µ 2 = 0 or, alternatively, H 0 : µ 1 = µ 2 The alternative hypothesis says what kind of difference we expect. H a : µ 1 - µ 2 > 0, H a : µ 1 - µ 2 < 0, or H a : µ 1 - µ 2 0 The Practice of Statistics, 5 th Edition 12

13 Significance Tests for µ 1 µ 2 To do a test, standardize x 1 - x 2 to get a two - sample t statistic : test statistic = statistic - parameter standard deviation of statistic t = (x 1 - x 2 ) - (m 1 - m 2 ) s 1 2 n 1 + s 2 2 n 2 To find the P-value, use the t distribution with degrees of freedom given by technology or by (df = smaller of n 1-1 and n 2-1). The Practice of Statistics, 5 th Edition 13

14 Significance Tests for µ 1 µ 2 Two-Sample t Test for the Difference Between Two Means The Practice of Statistics, 5 th Edition 14

15 Using Two-Sample t Procedures Wisely In planning a two-sample study, choose equal sample sizes if you can. Do not use pooled two-sample t procedures! We are safe using two-sample t procedures for comparing two means in a randomized experiment. Do not use two-sample t procedures on paired data! Beware of making inferences in the absence of randomization. The results may not be generalized to the larger population of interest. The Practice of Statistics, 5 th Edition 15

16 Comparing Two Means Section Summary In this section, we learned how to DESCRIBE the shape, center, and spread of the sampling distribution of the difference of two sample means. DETERMINE whether the conditions are met for doing inference about µ 1 µ 2. CONSTRUCT and INTERPRET a confidence interval to compare two means. PERFORM a significance test to compare two means. DETERMINE when it is appropriate to use two-sample t procedures versus paired t procedures. The Practice of Statistics, 5 th Edition 16

CHAPTER 10 Comparing Two Populations or Groups

CHAPTER 10 Comparing Two Populations or Groups 10. Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Means Learning