# STA291 Statistical Methods Lecture 23. Difference of means, redux Default case: assume no natural connection between individual observations in the two.

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STA291 Statistical Methods Lecture 23

Difference of means, redux Default case: assume no natural connection between individual observations in the two datasets. For example: compare average ROI for two investment firms May be difficult (impossible) or unnecessary in these instances to do pairing 2

Assumptions Two (simple) random samples, or randomization to two experimental groups Especially with small n, normality of sampled population(s) Independent samples 3

Completely Randomized Experimental Design Subjects (or, if not people, experimental units) are assigned/sampled at random 4

Notation 5 sample sizes for the first, second group sample means for the respective groups sample standard deviations for the groups standard error of the difference of the sample means

Hypothesis Testing Null Hypothesis: Test statistic: 6 H 0 :  1 –  2 =  0

Confidence Interval, but careful … 100 ( 1 –  )% Confidence Interval for  1 –  2 : where *Quick estimate of df = 7

Revisiting matched-pair … Paired data – natural connection between individual observations in the two data sets Has to be shown or included in information This setting allows for reduction of one source of variability when making an inference about any treatment variability (difference between parameters) Independent sample data Assumed situation Have to use standard error calculation 8

Example: Car Dealership Suppose you are the owner of a car dealership and you want to test average difference in discounts given to men and women. You take a random sample of 100 people and find of the 54 men an average discount of \$962.96 with standard deviation of \$458.95, and of the 46 women an average of \$1262.61with a standard deviation of \$399.70. Test the hypothesis that the average difference is equal to zero and give a 95% confidence interval for the difference.

Looking back o Two-sample problems in general o Difference of means: independent samples o Assumptions o Variance (P.T.o.S.) & standard error o Confidence intervals o Hypothesis testing o Difference from matched pair

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