Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 1 of 25 Chapter 11 Section 2 Inference about Two Means: Independent Samples

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 2 of 25 Chapter 11 – Section 2 ●Learning objectives  Test hypotheses regarding the difference of two independent means  Construct and interpret confidence intervals regarding the difference of two independent means 1 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 3 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 4 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 5 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 6 of 25 Chapter 11 – Section 2 ●Learning objectives  Test hypotheses regarding the difference of two independent means  Construct and interpret confidence intervals regarding the difference of two independent means 1 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 7 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 8 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 μ 1 and μ 2  The two sample sizes n 1 and n 2  The two sample means x 1 and x 2  The two sample standard deviations s 1 and s 2 ●We expect that x 1 – x 2 would be close to μ 1 – μ 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 9 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

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 10 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 (x 1 – x 2 ) – (μ 1 – μ 2 ) which has a standard deviation of approximately

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 11 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 has Student’s t-distribution with n – 1 degrees of freedom

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 12 of 25 Chapter 11 – Section 2 ●Thus for the test of two means, under certain appropriate conditions, the difference (x 1 – x 2 ) – (μ 1 – μ 2 ) is approximately Student’s t with mean 0, and the test statistic has an approximate Student’s t-distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 13 of 25 Chapter 11 – Section 2 ●This is Welch’s approximation, that has approximately a Student’s t-distribution ●The degrees of freedom is the smaller of  n 1 – 1 and  n 2 – 1

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 14 of 25 Chapter 11 – Section 2 ●For the particular case where be believe that the two population means are equal, or μ 1 = μ 2, and the two sample sizes are equal, or n 1 = n 2, then the test statistic becomes with n – 1 degrees of freedom, where n = n 1 = n 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 15 of 25 Chapter 11 – Section 2 ●Now for the overall structure of the test  Set up the hypotheses ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α ●Now for the overall structure of the test  Set up the hypotheses  Select the level of significance α  Compute the test statistic ●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 ●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 – 2 nd Edition – Chapter 11 Section 2 – Slide 16 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 n 1 and n 2 are at least 30) ●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 n 1 and n 2 are at least 30) ●These are the usual conditions we need to make our Student’s t calculations

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 17 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 18 of 25 Chapter 11 – Section 2 ●Compute the test statistic and the degrees of freedom, the smaller of n 1 – 1 and n 2 – 1 ●Compute the test statistic and the degrees of freedom, the smaller of n 1 – 1 and n 2 – 1 ●Compute the critical values (for the two-tailed, left-tailed, or right-tailed test)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 19 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 – 2 nd Edition – Chapter 11 Section 2 – Slide 20 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 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 μ 1 ≠ μ 2 to better illustrate the test statistic

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 21 of 25 Chapter 11 – Section 2 ●The test statistic is ●This has a Student’s t-distribution with 39 degrees of freedom ●The test statistic is ●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 ●The test statistic is ●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 – 2 nd Edition – Chapter 11 Section 2 – Slide 22 of 25 Chapter 11 – Section 2 ●Learning objectives  Test hypotheses regarding the difference of two independent means  Construct and interpret confidence intervals regarding the difference of two independent means 1 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 23 of 25 Chapter 11 – Section 2 ●Confidence intervals are of the form Point estimate ± margin of error ●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 x 1 – x 2  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 – 2 nd Edition – Chapter 11 Section 2 – Slide 24 of 25 Chapter 11 – Section 2 ●Thus confidence intervals are Point estimate ± margin of error Standard error Point estimate

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 2 – Slide 25 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