1. Two-Sample Inference Procedures with Means

2. Two independent samples Difference of Means

3. 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)

4. Confidence intervals: Called standard error Differences of Means (Using Independent Samples)

5. Degrees of Freedom Option 1: use the smaller of the two values n 1 – 1 and n 2 – 1 This will produce conservative results – higher p-values & lower confidence. Option 2: approximation used by technology Calculator does this automatically!

6. 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?

7. μ A = the true mean time it takes to commute taking Route A State the parameters Justify the confidence interval needed (state assumptions) 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. μ B = the true mean time it takes to commute taking Route B μ B - μ A = the true difference in means in time it takes to commute taking Route B f rom Route A Randomization- Assume two independent random samples of days

8. We are 95% confident that the true mean difference between the commute times is between -4.64 minutes and -1.36 minutes. Calculate the confidence interval. Explain the interval in the context of the problem. 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.

9. Hypothesis Statements: H 0 :  1 -  2 = hypothesized value  1 =  2 H a :  1 -  2 < hypothesized value  1 <  2 H a :  1 -  2 > hypothesized value  1 >  2 H a :  1 -  2 ≠ hypothesized value  1 ≠  2 Differences of Means (Using Independent Samples)

10. Hypothesis Test: State the degrees of freedom Differences of Means (Using Independent Samples)

11. 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: meanSDn Brand A20.18.712 Brand B18.97.512 Example 1 Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?

12. Parameters and Hypotheses μ A = the true mean absorption time in minutes for brand A Assumptions (Conditions) Since the conditions are met, a t-test for the two-sample means is appropriate. 1) Randomization- Assume two independent random samples H 0 : μ A - μ B = 0 H a : μ A - μ B  0 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. μ B = the true mean absorption time in minutes for brand B μ A - μ B = the true difference in means in absorption times in minutes for brands A and B

13. Calculations  = 0.05 Conclusion: Decision: Since p-value > , I fail to reject the null hypothesis at the.05 level. There is not sufficient evidence to suggest that these drugs differ in the speed at which they enter the blood stream

14. MC Answers 1)