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Two-Sample Inference Procedures with Means Suppose we have a population of adult men with a mean height of 71 inches and standard deviation of 2.6 inches. We also have a population of adult women with a mean height of 65 inches and standard deviation of 2.3 inches. Assume heights are normally distributed. Describe the distribution of the difference in heights between males and females (malefemale). Normal distribution with μx-y =6 inches & σx-y =3.471 inches Female 65 Male 71 Difference = male - female 6 Remember: x y x y x y 2 2 x y We will be interested in the difference of means, so we will use this to find standard error. Two-Sample Procedures with means • The goal of these inference procedures is to compare the responses to two treatments or to compare the characteristics of two populations. • We have INDEPENDENT samples from each treatment or population Assumptions: • Have two SRS’s from the populations or two randomly assigned treatment groups • Samples are independent • Both populations are normally distributed – Have large sample sizes – Graph BOTH sets of data • σ’s known/unknown Formulas Since in real-life, we will NOT know both σ’s, we will do t-procedures. Degrees of Freedom Option 1: use the smaller of the two values n1 – 1 and n2 – 1 This will produce conservative results – higher p-values & lower confidence. Calculator Option 2: approximation used bydoes this automatically! technology s s 2 2 1 2 1 2 2 n n df 1 s 1 s n 1 n n 1 n 1 2 2 1 2 1 2 2 Confidence Called intervals: standard error CI statistic critical value SD of statistic s s x x t * n n 1 2 2 1 2 1 2 2 Pooled procedures: • Used for two populations with the same variance • When you pool, you average the two-sample variances to estimate the common population variance. • DO NOT use on AP Exam!!!!! We do NOT know the variances of the population, so ALWAYS tell the calculator NO for pooling! Two competing headache remedies claim to give fastacting relief. An experiment was performed to compare the mean lengths of time required for bodily absorption of brand A and brand B. Assume the absorption time is normally distributed. Twelve people were randomly selected and given an oral dosage of brand A. Another 12 were randomly selected and given an equal dosage of brand B. The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow: mean SD n Brand A 20.1 8.7 12 Brand B 18.9 7.5 12 Describe the shape & standard error for sampling distribution of the differences in the mean speed of absorption. (answer on next screen) Describe the sampling distribution of the differences in the mean speed of absorption. Normal distribution with S.E. = 3.316 Find a 95% confidence interval difference in mean lengths of time required for bodily absorption of each brand. (answer on next screen) Assumptions: State assumptions! Have 2 independent from Think SRS “Price is volunteers Right”! Given the absorption is normally distributed Closestrate without going σ’s unknown over s12 s22 Formula & calculations x1 x2 t * df 21.53 n1 n2 2 2 8.7 7.5 20.1 18.9 2.080 (5.685,8.085) 12 12 Conclusion in context From calculator df = We are 95% confident that the true difference in mean 21.53, use t* for df = lengths of time required for bodily absorption of each 21 & 95% confidence brand is between –5.685 minutes and 8.085 minutes. level Note: confidence interval statements • Matched pairs – refer to “mean difference” • Two-Sample – refer to “difference of means” In a recent study on biofeedback, it was reported that meditation could alter the alpha & beta waves in the brain thus changing the rate at which the heart beats. This is important for relieving the effects of stress. Let’s test this! Hypothesis Statements: H H00:: μμ11 =- μμ22 = 0 H Haa:: Haa:: H Ha: μμ11<-μμ22 < 0 μμ11>-μμ22 > 0 μ1 -≠ μμ22 ≠ 0 Be sure to define BOTH 1 and 2! Hypothesis Test: Test statistic Since we usually assume H0 is true, statistic parameter then this equals 0 – can usually SDsoofwestatistic leave it out x x t 1 2 1 2 2 1 2 1 2 s s n n 2 The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow: Brand A Brand B mean 20.1 18.9 SD 8.7 7.5 n 12 12 Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream? Assump.: Have 2 independent SRS from volunteers State assumptions! Given the absorption rate is normally distributed σ’s unknown H0: μA= μB Hypotheses & define variables! Where μA is the true mean absorption time for Brand A & μB is the true mean absorption time for Brand B Ha:μA= μB x1 - x2 20.1 - 18.9 t= = = .361 Formula & calculations 2 2 2 2 s1 s2 8.7 7.5 + + n1 n2 12 12 Conclusion in context p - value = .7210 df = 21.53 a = .05 Since p-value > α, I fail to reject H0. There is not sufficient evidence to suggest that these drugs differ in the speed at which they enter the blood stream. Suppose that the sample mean of Brand B is 16.5, then is Brand B faster? t x1 x2 s12 s22 n1 n2 20.1 16.5 8.7 2 7.52 12 12 1.085 p value .2896 df 21.53 α .05 No, I would still fail to reject the null hypothesis. Robustness: • Two-sample procedures are more robust than one-sample procedures • BEST to have equal sample sizes! (but not necessary) A modification has been made to the process for producing a certain type of time-zero film (film that begins to develop as soon as the picture is taken). Because the modification involves extra cost, it will be incorporated only if sample data indicate that the modification decreases true average development time by more than 1 second. Should the company incorporate the modification? Original 8.6 5.1 4.5 5.4 Modified 5.5 4.0 3.8 6.0 6.3 6.6 5.8 4.9 5.7 8.5 7.0 5.7