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Section 13.1 COMPARING TWO MEANS What’s the difference between what we did “yesterday” and what we are doing today???? In the previous chapter we were performing inference on a single population. Now we are comparing two populations or two treatments. This is a much more common objective in the practice of statistics. Pulse Rate Experiment Choose a card Record the letter from the card on a piece of paper Calculate your pulse rate (beats per min.) Record your pulse rate on the same piece of paper Teacher will calculate average rates for the two groups DID YOU KNOW? The letter on the card influences your pulse rate! Pulse Rate Experiment Continued Don’t believe it? Look at the differences and you can see that the two groups are different. Am I right? How can we tell? Assumptions for Inference when Comparing Two Means 1. 2. 3. The data comes from TWO SRSs from TWO distinct populations. Observations from both populations have a normal distribution with unknown means (1 & 2) and unknown standard deviations (σ1 & σ2 ). While this is ideal, in practice, it is enough that the distributions have similar shapes and that there are no outliers. The samples should be independent of one another. When sampling without replacement, we also want each population to be at least 10 times as big as its respective sample. z Two-Sample z Statistic vs. Two-Sample t Statistic x1 x2 ( 1 2 ) 2 1 n1 t 2 2 n2 x1 x2 ( 1 2 ) 2 1 NOTE : SEx1 x2 2 2 s s n1 n2 x1 x2 2 1 2 2 s s n1 n2 s12 s22 n1 n2 Two-Sample t Procedures The two-sample t statistic DOES NOT, in general, have a t distribution. These procedures always err on the safe side, reporting HIGHER P-values and LOWER confidence than may actually be true. The gap between what is reported and the truth is quite small unless the samples sizes are both small and unequal. ROBUSTNESS Recall, one-sample t procedures were robust against the Normality assumption provided there were no outliers or extreme skewness The two-sample t procedures are more robust than the one-sample t methods, particularly when the distributions are not symmetric. This is particularly true when the samples are the same size and the two population distributions have similar shape. (even for samples as small as 5) Therefore . . . This is when we use the two-sample t procedures Remember, it is best for samples sizes to be equal so ALWAYS choose equal samples sizes if you can. It’s more important for the data to be SRSs from the populations than the populations have normal distributions If n1 + n2 < 15, the data must be normal to use tprocedures If n1 + n2 ≥ 15, the t-procedures can be used except if there are outliers or strong skewness If n1 + n2 ≥40, t-procedures can be used even in the presence of strong skewness (outliers still bad) t Statistic 2 1 2 2 s s n1 n2 The statistic does not have a normal distribution IT ALSO DOES NOT HAVE A t-DISTRIBUTION Degrees of freedom: TWO METHODS x1 x2 1 2 t smaller of n1-1 and n2-1 (this gives more conservative results) complex formula (which yields very accurate approximations) Recall, when subtracting two random variables, the variances still add t statistic says how far x1 x2 is from its mean 1 2 in standard deviation units Degrees of Freedom There are two methods, both of which are conservative enough that the true P-value or fixed significance level will always be equal to or less than the value calculated from t(k) More Conservative Method df=the smaller of n1-1 and n2-1 Less Conservative Method (using technology) This formula is what is used by technology such as your calculator to obtain a more accurate, but still conservative 2 result 2 2 s1 s2 n1 n2 df 2 2 2 2 1 s1 1 s2 n1 1 n1 n2 1 n2 Confidence Intervals x1 x2 ± t* 2 1 2 2 s s n1 n2 t* is the upper (1-C)/2 critical value for the t(k) distribution with k being the smaller of n1-1 and n2-1 (unless using technology) Recall, this will yield a conservative interval The Steps for a TWO-SAMPLE t-TEST State the hypothesis and name test 1. Ho: 1 = 2 (You may see this as 1- 2 = 0) Ha: 1 ‹, ›, or ≠ (You may see this as 1- 2 ‹, ›, or ≠ 0) State and verify your assumptions Calculate the P value and other important values 2. 3. - Done in calculator or… Book: - 4. 2 Using Table C, look in the df (k) column and then look across the line to find the range of probabilities the t statistic falls in State Conclusions (Both statistically and contextually) - The smaller the p-value, the greater the evidence is to reject Ho When should we use the Pooled Two-Sample t Procedures? NEVER! Always choose NO when asked. Pooling can only be justified when the two populations have the same variance. The advantage of pooling is that it has exactly the t distribution with n1 + n2 – 2 degrees of freedom if the two variances really are equal. They usually aren’t. Example: Effects of light on plant growth In an experiment to study the effect of the spectrum of the ambient light on the growth of plants, researchers assigned tobacco seedlines at random to two groups of eight plants each. The plants were grown in a greenhouse under identical conditions except for lighting. The experimental group was grown under blue light, the control group under natural light. Here is the data on stem growth (in mm). Control Group 4.3 4.2 3.9 4.1 4.1 4.2 3.8 4.1 (Natural Light) Experimental 3.1 2.9 3.2 3.2 2.7 2.9 3.0 3.1 (Blue Light) Construct and interpret a 90% confidence interval for the difference in mean plant growth. Step 1—Parameters This is a two-sample situation. Call plants grown under natural light Population 1 and plants grown under blue light Population 2. Then the corresponding parameters are 1= mean stem growth of tobacco plants under natural light and 2 = mean stem growth of tobacco plants under blue light. We want to estimate the amount by which blue light reduces stem growth during this period, 1-2 Step 2—Conditions We will construct a two-sample t interval for the difference in means if the conditions are satisfied. SRS—The data were produced in a randomized comparative experiment, so we should be able to attribute any significant difference in means to the difference in treatments. We do not know if these seedlings are representative of the population of all seedlings of this type of plant, which may limit our ability to generalize. Normality—We also need to know that the two population distributions are approximately Normal. Boxplots and Normal probability plots of the sample data give us no reason to doubt the validity of this condition. Independence—The random assignment helps ensure that we have two independent samples of growth measurements. Step 3—Calculations (more conservative) Population 1 2 Sample Size 8 8 Sample Mean 4.0875 3.0125 Sample Variance 0.0270 0.0298 There are 7 degrees of freedom, because n1-1 and n2-1 are 7. The 90% confidence interval with df=7 uses t* = 1.895. The resulting confidence interval for the difference 1 - 2 between the population means is s12 s22 0.0270 0.0298 ( x1 x2 ) t * (4.09 3.01) 1.895 n1 n2 8 8 1.08 0.16 (0.92,1.24) Step 3—Calculations (less conservative) Population 1 2 Sample Size 8 8 Sample Mean 4.0875 3.0125 Sample Variance 0.0270 0.0298 There are 13.9646 degrees of freedom based on the complex formula used by the calculator. The 90% confidence interval with df=13.9646 uses t* = 1.7616. The resulting confidence interval for the difference 1 - 2 between the population means is s12 s22 0.0270 0.0298 ( x1 x2 ) t * (4.09 3.01) 1.7616 n1 n2 8 8 (0.92658,1.2234) from calculator Step 4—Interpretation We are 90% confident that the reduction in average stem growth due to the blue light is between 0.92 and 1.24 millimeters. We have this confidence because if we were to repeat this process over and over again, 90% of all created intervals would capture the true difference in growth. A two-sample t Test on the same example information Parameters: Ho: 1 = 2 Null hyp. is that tobacco plants will not grow any differently under blue light versus natural light. Alt. hyp. is that the light will impact growth. P-value=.000000004 df = 13.9646 Ha: 1 ≠ 2 t x1 x2 2 1 2 2 12.7595 s s n1 n2 We can confidently reject Ho due to the extremely low P-value (if truly no difference, nearly no chance of seeing difference of 1.08) Tobacco plants appear to have a distinctly different growth rate in blue light versus natural light.