1. Two-Sample Inference Procedures with Means

2. Two-Sample Procedures with Means
Goal: Compare two different populations/treatments
INDEPENDENT samples from each population/treatment

3. Remember: When combining two random variables X and Y,
This formula only works if X and Y are independent

4. μM-F = 6 inches & σM-F = 3.471 inches
Suppose we have a population of adult
men with a mean height of 71 inches
and standard deviation 2.6 inches.
We also have a population of adult women with a mean height of 65 inches and standard deviation 2.3 inches. Heights are normally distributed.
Describe the distribution of the difference in heights between males and females (male –female).
Normal distribution
μM-F = 6 inches & σM-F = inches

5. 71 65
Female
Male 6
Difference (male – female)
σ = 3.471

6. What is the probability that a randomly selected man is at most 5 inches taller than a randomly selected woman?
b) What is the 70th percentile for the difference (male – female) in heights of a randomly selected man and woman?
P(xM – xF < 5) = normalcdf(-1E99, 5, 6, 3.471)
= .3866
(xM – xF) = invNorm(.7, 6, 3.471) = 7.82

7. Calculator Simulation!
a) What is the probability that the mean height of 30 men is at most 5 inches taller than the mean height of 30 women?
b) What is the 70th percentile for the difference (male – female) in mean heights of 30 men and 30 women?
P(xM – xW < 5) = .057
6.332 inches

8. Conditions for Two Means
Two independent SRS's (or randomly assigned treatments)
Both samp. dist. are approx. normal
Both populations normal
Both n's > 30
Both graphs linear

9. Calculator does this automatically!
Degrees of Freedom
Option 1: Use the smaller df: n1 – 1 or n2 – 1
 Using the larger one overestimates the collective sample sizes
Option 2: Welch-Satterthwaite approximation
Calculator does this automatically!

10. Confidence Interval for the Difference of Two Means
Standard Error/ Deviation

11. Two competing headache remedies claim to give fast-acting relief
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. Absorption time is normally distributed. Twelve people were randomly selected and given a 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

12. a) Describe the sampling distribution of the differences in the mean speed of absorption (A – B).
b) Construct a 95% confidence interval for the difference in mean lengths of time (A – B) required for bodily absorption of each brand.
Normal; s = 3.316
Conditions:
2 independent randomly assigned treatments
Populations are normal

13. Think “Price is Right”: Closest df without going over
From calculator
We are 95% confident that the true difference in mean absorption time (A minus B) is between minutes and minutes.
If we made lots of intervals this way, 95% of them would contain the true difference in means.

14. A Subtle Distinction Matched pairs: “mean difference”
Two-sample inference: “difference in means”

15. Hypothesis Statements
Ha: μ1 – μ2 < 0
Ha: μ1 – μ2 > 0
Ha: μ1 – μ2 ≠ 0
Be sure to define BOTH μ1 and μ2!
Ha: μ1 < μ2
Ha: μ1 > μ2
Ha: μ1 ≠ μ2

16. Test Statistic
Since we assume H0 is true, this part equals 0 – so we can leave it out

17. c) Is there sufficient evidence that the two brands differ in the speed at which they enter the bloodstream?
Conditions:
2 independent randomly assigned treatments
Populations are normal

18. H0: mA= mB
Ha: mA ≠ mB
Where μA and μB are the true mean absorption times
p-value = α = .05
Since p-value > α, we fail to reject H0. There is not sufficient evidence to suggest these drugs differ in their absorption time.

19. Pooling Used for two populations with the same variance (σ2)
Pooling = Averaging the two s2 to estimate σ2
We almost never pool for means, since we don't know σ

20. Robustness
Two-sample procedures: more robust than one-sample procedures
Most robust with equal sample sizes (but not necessary!)

21. A modification has been made to the process for producing a certain type of film. Since the modification costs extra, it will be incorporated only if sample data indicate that the modification decreases the true average development time. At a significance level of 10%, should the company incorporate the modification?
Original
Modified
Conditions:
2 independent SRS's of film
Normal prob. plots linear  approx. normal sampling dist.’s

22. Where μO and μM are the true mean developing times for original and modified film
H0: μO = μM
Ha: μO > μM
p-value = α = .1
Since p-value < α, we reject H0. There is sufficient evidence to suggest the company should incorporate the modification.