1. Two independent samples Difference of Means

2. 1) The two samples are chosen independently of each other. OR
Differences of Means (Using Independent Samples)
CONDITIONS:
1) The two samples are chosen independently of each other. OR
The treatments are randomly assigned to individuals or objects or (vice versa)
2) 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
3) 10% rule – Both samples should be less than 10% of their respective populations

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

4. Calculator does this automatically!
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.
Option 2: approximation used by technology
Calculator does this automatically!

5. Hypothesis Statements:
Differences of Means (Using Independent Samples)
Hypothesis Statements:
H0: m1 - m2 = hypothesized value
Ha: m1 - m2 < hypothesized value
Ha: m1 - m2 > hypothesized value
Ha: m1 - m2 ≠ hypothesized value
H0: m1 = m2
Ha: m1< m2
Ha: m1> m2
Ha: m1 ≠ m2

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

7. 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: mean SD n Brand A Brand B
Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?

8. 4) A and B are both unknown
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) The samples must be random and independent which is stated in the problem.
2) The sample distributions should be approximately normal. Since it is stated in the problem that the population is normal then the sample distributions are aprroximately normal.
3) The samples should be less than 10% of their populations. The population should be at least 240 people, which I will assume.
4) A and B are both unknown
Since the conditions are met, a t-test for the two-sample means is appropriate.

9. Calculations
 = 0.05

10. Decision:
Since p-value > a, I fail to reject the null hypothesis at the .05 level.
Conclusion:
There is not sufficient evidence to suggest that there is a difference in the true mean absorption time in minutes for Brand A and Brand B.

11. #5 Commuting.
A man who moves to a new city sees that there are two routes 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 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 a standard deviation of 3 minutes, and Route B takes an average of 43 minutes, with a standard deviation of 2 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers.
Find a 95% confidence interval for the difference in average commuting time for the two routes.
Should the man believe the old-timer’s claim that he can save an average of 5 minutes a day by always driving Route A? Explain.

12. Justify the confidence interval needed (state assumptions)
Page 567: #5
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 from Route A
Justify the confidence interval needed (state assumptions)
1) The samples must be random and independent which is state in the problem.
2) 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.
3) The samples should be less than 10% of the population. The population should be at least 400 days, which I will assume.
4) A and B are both unknown
Since the conditions are satisfied a t – interval for the difference of means is appropriate.

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