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Two-Sample Inference Procedures with Means Two independent samples Difference of Means Differences of Means (Using Independent Samples) CONDITIONS: 1) We have 2 SRS from 2 distinct populations 2) Both samples are chosen independently OR the treatments are randomly assigned to individuals or objects 3) 10% rule – Both samples should be less than 10% of their respective populations 4) The sample distributions for both samples should be approximately normal - the populations are known to be normal, or - the sample sizes are large (n 30), or - graph data to show approximately normal Differences of Means (Using Independent Samples) 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 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 Ex1. A man who moves to a new city sees that there are 2 routes that he could take to work. A neighbor who has lived there a long time tells him Route A will average 5 minutes faster than Route B. The man decides to do an experiment. Each day he flips a coin to determine which way to go, driving each route 20 days. He finds that Route A takes an average of 40 minutes with standard deviation 3 minutes, and Route B takes an average of 43 minutes with standard deviation 2 minutes. His histogram of travel times are roughly symmetric and show no outliers. Find a 95% confidence interval for the difference in the average commuting time for the 2 routes. Should the man believe the neighbor’s claim that he can save an average of 5 minutes by driving Route A? State the parameters μA = the true mean time it takes to commute taking Route A μB = the true mean time it takes to commute taking Route B μB - μA = the true difference in means in time it takes to commute taking Route B f rom Route A Justify the confidence interval needed (state assumptions) Randomization- Assume two independent random samples of days 10%-The samples should be less than 10% of the populations. The populations should be at least 200 days for each route, which I will assume. Nearly normal- The sample distributions should be approximately normal. It is stated in the problem that graphs of the travel times are roughly symmetric and show no outliers, so we will assume the distributions are approximately normal. Since the conditions are satisfied a t – interval for the difference of means is appropriate. Calculate the confidence interval. xA xB t 22 32 s A2 sB2 4.64, 1.36 (40 43) 2.034 20 20 nA nB in vT (.975,33.1) df 33.1 (40 43) 2.09 in vT (.975,19) 22 32 (4.687, 1.313) 20 20 df 19 Explain the interval in the context of the problem. We are 95% confident that the true mean difference between the commute times is between -4.64 minutes and -1.36 minutes. The man should not believe the neighbor’s claim that he can save 5 minutes since based on the interval he would only save between 1.36 and 4.64 minutes. Differences of Means (Using Independent Samples) Hypothesis Statements: H0: m1 - m2 = hypothesized value m1 = m2 Ha: m1 - m2 < hypothesized value m1 < m2 Ha: m1 - m2 > hypothesized value m1 > m2 Ha: m1 - m2 ≠ hypothesized value m1 ≠ m2 Differences of Means (Using Independent Samples) Hypothesis Test: statistic - parameter Test statistic SD of statistic x x m m t State the degrees of freedom 1 2 1 2 2 1 2 1 2 s s n n 2 Example 1 Two competing headache remedies claim to give fast-acting 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: 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? Parameters and Hypotheses μA = the true mean absorption time in minutes for brand A μB = the true mean absorption time in minutes for brand B μA - μB = the true difference in means in absorption times in minutes for brands A and B H0: μA - μB = 0 Ha: μA - μB 0 Assumptions (Conditions) 1) Randomization- Assume two independent random samples 2) 10% - The samples should be less than 10% of their populations. The populations should be at least 120 people for each drug, which I’ll assume. 3) Nearly normal- The sample distributions should be approximately normal. Since it is stated in the problem that the population is normal then the sample distributions are approximately normal. Since the conditions are met, a t-test for the two-sample means is appropriate. Calculations = 0.05 xA xB m A mB 20.1 18.9 0 .3619 t s A2 sB2 nA nB 8.7 2 7.52 12 12 p value 2 P(t .3619) .721 df 21.5 p value 2 P(t .3619) .724 df 11 Decision: Since p-value > , I fail to reject the null hypothesis at the .05 level. Conclusion: There is not sufficient evidence to suggest that these drugs differ in the speed at which they enter the blood stream MC Answers 1)