Two Independent Samples Question In 2000, did men and women differ in terms of their body mass index? We want to extend our inquiry into data by looking at the comparison of two groups.

Populations 1. Female 2. Male Inference random selection random Samples

Body Mass Index Females Males Work with the sample statistics. Remember to pool the information about the error standard deviation from the two separate samples into one value.

95% Confidence Interval

95% Confidence Interval Note that the table in your text book does not have 98 degrees of freedom. I used JMP to get the t* value.

Interpretation We are 95% confident that the difference in population mean BMI for women compared to men is between –2.38 and 3.61. Women could have a mean BMI as much as 2.38 lower than men or as much as 3.61 higher than men.

Difference? Because zero is in the confidence interval, there could be no difference in population mean BMI for women compared to men. This agrees with the test of hypothesis.

Two-sample model Y represents a value of the variable of interest represents the ith population mean represents the random error associated with an observation For each individual in our population we can model the value of the variable of interest (in our example BMI) as being a population mean value plus some random error (the random error could be a positive or negative values). Because we have males and females there is a population mean value for males and a separate population mean value for females.

Conditions The random error term, , is Independent Identically distributed Normally distributed with standard deviation, These are the usual normal model conditions mentioned in Stat 101. They are really no different from the one-sample model conditons.

Residuals Estimate of error (Observation – Fit) Residual A residual is an estimate of the error term for an individual observation. As such it is simply the difference between the observation and the fitted model parameter. Because we have two separate groups, residuals are calculated separately for each group but then combined into one set of residuals for checking the conditions.

Checking Conditions Independence. Hard to check this but the fact that we obtained the data through separate random samples of women and men assures us that the statistical methods should work.

Checking Conditions Identically distributed. Check using an outlier box plot. Unusual points may come from a different distribution Check using a histogram. Bi-modal shape could indicate two different distributions. The outlier box plot sets up “fences” beyond which individual values are considered unusual when compared to the rest of the sample.

Checking Conditions Normally distributed. Check with a histogram. Symmetric and mounded in the middle. Check with a normal quantile plot. Points falling close to a diagonal line. Histograms can be misleading. Different groupings (bar placements) can different impressions. Do not always rely on the default histogram given to you by JMP. You may have to fool with the horizontal axis settings or use the grabber tool. The normal quantile plot is a more reliable means of assessing whether the sample could have come from a normal distribution.

Residuals from the BMI data Residuals from the BMI data. For women, take each observed BMI and subtract off the mean for women. For men, take each observed BMI and subtract off the mean for men. Combine all 100 residuals into one set for analysis.

BMI Residuals Histogram is skewed left and mounded to the right of zero. Box plot is fairly symmetric with two potential outliers on the high side. Normal quantile plot has points following the diagonal line for the first part but then wiggles around for larger values.

BMI Residuals The conditions for statistical inference may not be met for these data.

Consequences The P-value for the test may not be correct. Even so, there is not much of a difference between women and men, and I would not change my conclusion from the test of hypthesis.

Consequences The stated confidence level may not give the true coverage rate. I would still use the confidence interval but recognize that the true coverage rate is probably less than 95%.