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Chapter 11 Section 2 Inference about Two Means: Independent Samples Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 1 of 25 Chapter 11 – Section 2 ● Learning objectives 1 Test hypotheses regarding the difference of two independent means 2 Construct and interpret confidence intervals regarding the difference of two independent means Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 2 of 25 Chapter 11 – Section 2 ● Two samples are independent if the values in one have no relation to the values in the other ● Examples of not independent Data from male students versus data from business majors (an overlap in populations) The mean amount of rain, per day, reported in two weather stations in neighboring towns (likely to rain in both places) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 3 of 25 Chapter 11 – Section 2 ● A typical example of an independent samples test is to test whether a new drug, Drug N, lowers cholesterol levels more than the current drug, Drug C ● A group of 100 patients could be chosen The group could be divided into two groups of 50 using a random method If we use a random method (such as a simple random sample of 50 out of the 100 patients), then the two groups would be independent Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 4 of 25 Chapter 11 – Section 2 ● The test of two independent samples is very similar, in process, to the test of a population mean ● The only major difference is that a different test statistic is used ● We will discuss the new test statistic through an analogy with the hypothesis test of one mean Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 5 of 25 Chapter 11 – Section 2 ● Learning objectives 1 Test hypotheses regarding the difference of two independent means 2 Construct and interpret confidence intervals regarding the difference of two independent means Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 6 of 25 Chapter 11 – Section 2 ● For the test of one mean, we have the variables The hypothesized mean (μ) The sample size (n) The sample mean (x) The sample standard deviation (s) ● We expect that x would be close to μ Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 7 of 25 Chapter 11 – Section 2 ● In the test of two means, we have two values for each variable – one for each of the two samples The two hypothesized means μ1 and μ2 The two sample sizes n1 and n2 The two sample means x1 and x2 The two sample standard deviations s1 and s2 ● We expect that x1 – x2 would be close to μ1 – μ2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 8 of 25 Chapter 11 – Section 2 ● For the test of one mean, to measure the deviation from the null hypothesis, it is logical to take x–μ which has a standard deviation of approximately s2 n Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 9 of 25 Chapter 11 – Section 2 ● For the test of two means, to measure the deviation from the null hypothesis, it is logical to take (x1 – x2) – (μ1 – μ2) which has a standard deviation of approximately s12 s22 n1 n2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 10 of 25 Chapter 11 – Section 2 ● For the test of one mean, under certain appropriate conditions, the difference x–μ is Student’s t with mean 0, and the test statistic t x s2 n has Student’s t-distribution with n – 1 degrees of freedom Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 11 of 25 Chapter 11 – Section 2 ● Thus for the test of two means, under certain appropriate conditions, the difference (x1 – x2) – (μ1 – μ2) is approximately Student’s t with mean 0, and the test statistic t ( x1 x 2) ( 1 2 ) s12 s22 n1 n2 has an approximate Student’s t-distribution Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 12 of 25 Chapter 11 – Section 2 ● This is Welch’s approximation, that t ( x1 x 2) ( 1 2 ) s12 s22 n1 n2 has approximately a Student’s t-distribution ● The degrees of freedom is the smaller of n1 – 1 and n2 – 1 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 13 of 25 Chapter 11 – Section 2 ● For the particular case where be believe that the two population means are equal, or μ1 = μ2, and the two sample sizes are equal, or n1 = n2, then the test statistic becomes t ( x1 x 2 ) s2 s2 2 1 n with n – 1 degrees of freedom, where n = n1 = n2 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 14 of 25 Chapter 11 – Section 2 ● Now for the overall structure of the test Set up the hypotheses Select the level of significance α Compute the test statistic Compare the test statistic with the appropriate critical values Reach a do not reject or reject the null hypothesis conclusion Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 15 of 25 Chapter 11 – Section 2 ● In order for this method to be used, the data must meet certain conditions Both samples are obtained using simple random sampling The samples are independent The populations are normally distributed, or the sample sizes are large (both n1 and n2 are at least 30) ● These are the usual conditions we need to make our Student’s t calculations Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 16 of 25 Chapter 11 – Section 2 ● State our two-tailed, left-tailed, or right-tailed hypotheses ● State our level of significance α, often 0.10, 0.05, or 0.01 Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 17 of 25 Chapter 11 – Section 2 ● Compute the test statistic t ( x1 x 2) ( 1 2 ) s12 s22 n1 n2 and the degrees of freedom, the smaller of n1 – 1 and n2 – 1 ● Compute the critical values (for the two-tailed, left-tailed, or right-tailed test) Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 18 of 25 Chapter 11 – Section 2 ● Each of the types of tests can be solved using either the classical or the P-value approach ● Based on either of these methods, do not reject or reject the null hypothesis Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 19 of 25 Chapter 11 – Section 2 ● We have two independent samples The first sample of n = 40 items has a sample mean of 7.8 and a sample standard deviation of 3.3 The second sample of n = 50 items has a sample mean of 11.6 and a sample standard deviation of 2.6 We believe that the mean of the second population is exactly 4.0 larger than the mean of the first population We use a level of significance α = .05 ● We use an example with μ1 ≠ μ2 to better illustrate the test statistic Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 20 of 25 Chapter 11 – Section 2 ● The test statistic is t ( x1 x 2) ( 1 2 ) s12 s22 n1 n2 ( 7.8 12.9 ) 4.0 3.32 2.62 40 50 1.72 ● This has a Student’s t-distribution with 39 degrees of freedom ● The two-tailed critical value is 2.02, so we do not reject the null hypothesis ● We do not have sufficient evidence to state that the deviation from 4.0 is significant Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 21 of 25 Chapter 11 – Section 2 ● Learning objectives 1 Test hypotheses regarding the difference of two independent means 2 Construct and interpret confidence intervals regarding the difference of two independent means Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 22 of 25 Chapter 11 – Section 2 ● Confidence intervals are of the form Point estimate ± margin of error ● We can compare our confidence interval with the test statistic from our hypothesis test The point estimate is x1 – x2 We use the denominator of the test statistic (Welch’s approximation) as the standard error We use critical values from the Student’s t Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 23 of 25 Chapter 11 – Section 2 ● Thus confidence intervals are Point estimate ± margin of error ( x1 x2 ) t / 2 s12 s22 n1 n2 Point estimate Standard error Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 24 of 25 Summary: Chapter 11 – Section 2 ● Two sets of data are independent when observations in one have no affect on observations in the other ● In this case, the differences of the two means should be used in a Student’s t-test ● The overall process, other than the formula for the standard error, are the general hypothesis test and confidence intervals process Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 11 Section 2 – Slide 25 of 25