2t methods

Two Samples, Averages: 

Two Samples, Averages Two sample T tests and confidence intervals

Data Situation: 

Data Situation There are now two populations denoted one and two. We are typically interested in how these two population means compare. We denote the mean in population one, mu_1 and the mean for population 2, mu_2.

Sample Data: 

Sample Data

Confidence Interval Formula: 

Confidence Interval Formula

Degrees of Freedom: 

Degrees of Freedom The degrees of freedom, df for a two-sample test or CI depend on how the analysis is performed. If the data are being analyzed with a computer package, use the degrees of freedom supplied in the output. If the data are being analyzed “by-hand” on a quiz or exam, then use df=min(n1-1, n2-1).

Example: 

Example If one sample has 12 observations and the other has 19. The degrees of freedom used by hand would be min(12-1, 19-1) = min(11,18)=11.

Hypothesis Test Formula: 

Hypothesis Test Formula

Assumptions for Using T Procedures: 

Assumptions for Using T Procedures T tests and confidence intervals are technically only appropriate IF the original data distributions are independent and normally distributed. If n_1=n_2, and the shapes of the distributions are similar, p-values and CI’s will be accurate for sample sizes as low as n_1=n_2=5. That is, we get extra safety from having similar sample sizes in the two groups.

T-Procedure Assumptions: 

T-Procedure Assumptions If n1+n2<15, use t-procedures if the data are close to normally distributed. Don’t use t if the data have skewness or outliers. Use a non-parametric method, or transform data. If n1+n2 is larger than 15, but less than 40, use T methods except in the presence of outliers or strong skewness. If n1+n2 > 40, large samples, can use t-procedures even for skewed distributions.

Hot Dogs: 

Hot Dogs Mystery Meat Dogs: n=17, x-bar 158.706, s=25.236 Poultry Dogs: n=17, x-bar=122.471, s=25.483 Measurements are calories per dog. Are the average calorie contents different?

Hypothesis Test: Hot Dogs: 

Hypothesis Test: Hot Dogs

Conclusion: Hot Dogs: 

Conclusion: Hot Dogs The data is unlikely to occur if the Ho is true, so the data are inconsistent with Ho. There is evidence to doubt the Ho. There is evidence to support the Ha. There is evidence the mean calorie contents differ. Because the sample mean for mystery meat is greater than the poultry dogs, there is evidence the Mys Meat dogs have more cals on average.

Confidence Interval: Hot Dogs: 

Confidence Interval: Hot Dogs

CI Conclusion: Hot Dogs: 

CI Conclusion: Hot Dogs The plausible range of values for the difference of mu_mm minus mu_poultry is (17.79, 54.67). This range is entirely above zero. This is evidence the difference in the means is positive, and this means evidence the mystery meat dogs have a higher average cal content than poultry dogs.

Hot Dog Display: 

Hot Dog Display

Male Height Example: 

Male Height Example Reference: Nature, volume 403, no 6766, p 156, (2000). Pawlowski et al. Study of marriage and children of Polish men in their 20’s and 50’s. Data collect in late 80’s. 20’s: 942 with no children, 217 with children 50’s: 141 with no children, 268 with children Is height of males related to marriage status or number of children fathered?

Male Height Example: 

Male Height Example For males in their 20’s, the hypotheses are: Ho: mu_no – mu_with = 0 Ha: mu_no – mu_with != 0 Null hypothesis says average height of males without children is same as for those with children. Two sample t-statistic = -2.97 P-Value= Small (why?)

Male Height Example: 

Male Height Example Conclusion for men in their 20’s is that there is evidence that males with children are sig taller than those without children. What about men in their 50’s?

Male Height Example: 

Male Height Example Same hypotheses T-statistic=.17 P-Value= Big (Why?) Conclusion here is that there is no evidence the mean heights differ for those with and without children. Why do men in 20’s have different pattern than those in 50’s?

Male Height Example: 

Male Height Example Answer: Ever heard of WW II? Men in their 50’s in the 80’s were in their 10’s and 20’s in the 1940’s (prime breeding time !!!). Remember that these are Polish men, and were nearly eliminated by Germany in WW II. The researchers believe that there was such competition among women for mates that attributes like height were unimportant. However in modern times, Polish women can be selective and tend to select taller men for mates given the opportunity. I have no comment on this theory (;-) !

Male Height Example: CI’s: 

Male Height Example: CI’s The two sample confidence interval for men in their 20’s would have had the form: (-, -), and thus evidence the difference in means mu_no – mu_with was <0, and evidence the average height was greater for those with children. For those in 50’s, the CI would have been of form (-,+), and thus conclude no evidence the mean heights differ.