Two-Sample Inference Procedures with Means

Remember: We will be intereste d in the differen ce of means, so we will use this to find standard error.

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 (male- female). Normal distribution with  x-y =6 inches &  x-y =3.471 inches

7165 Female Male 6 Difference = male - female  = 3.471

a)What is the probability that the height of a randomly selected man is at most 5 inches taller than the height of a randomly selected woman? b) What is the 70th percentile for the difference (male-female) in heights of a randomly selected man & woman? P((x M -x F ) < 5) = normalcdf(-∞,5,6,3.471) =.3866 (x M -x F ) = invNorm(.7,6,3.471) = 7.82

Do 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?

Two-Sample Procedures with means two treatments two populationsThe 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 When we compare, what are we interested in?

Assumptions: two SRS’s two randomly assignedHave two SRS’s from the populations or two randomly assigned treatment groups Samples are independent Both distributions are approximately normal –Have large sample sizes –Graph BOTH sets of data  ’s  ’s unknown

Formulas NOT Since in real-life, we will NOT know both  ’s, we will do t-procedures.

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!

Confidence intervals: Called standard error

Pooled procedures: sameUsed 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 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 A Brand B 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. Find a 95% confidence interval difference in mean lengths of time required for bodily absorption of each brand. Normal distribution with S.E. = 3.316

Note: confidence interval statements “mean difference”Matched pairs – refer to “mean difference” “difference of means”Two-Sample – refer to “difference of means”

Hypothesis Statements: H 0 :  1 -  2 = 0 H a :  1 <  2 < 0 H a :  1 >  2 > 0 H a :  1 ≠  2 ≠ 0 H 0 :  1 =  2 BOTH Be sure to define BOTH  1 and  2 !

Hypothesis Test: Since we usually assume H 0 is true, then this equals 0 – so we can usually leave it out

The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The results follow: meanSDn Brand A Brand B Is there sufficient evidence that these drugs differ in the speed at which they enter the blood stream?

Have 2 independent randomly assigned treatments Given the absorption rate is normally distributed  ’s unknown Since p-value > a, I fail to reject H 0. There is not sufficient evidence to suggest that these drugs differ in the speed at which they enter the blood stream. State assumptions! Formula & calculations Conclusion in context H 0 :  A =  B H a :  A =  B Where  A is the true mean absorption time for Brand A &  B is the true mean absorption time for Brand B Hypotheses & define variables! On Calc: Go to Stat – Test Input Statistics No Pooling Calculate

Suppose that the sample mean of Brand B is 16.5, then is Brand B faster? No, I would still fail to reject the null hypothesis.

Robustness: more robustTwo-sample procedures are more robust than one-sample procedures BESTBEST 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 Modified

Assume we have 2 independent SRS of film Both distributions are approximately normal due to approximately symmetrical boxplots  ’s unknown H 0 :  O -  M = 1 H a :  O -  M > 1 Where  O is the true mean developing time for original film &  M is the true mean developing time for modified film Since p-value > , I fail to reject H 0. There is not sufficient evidence to suggest that the company incorporate the modification.