Exam 13: Asking and Answering Questions About the Difference Between Two Means

After an outbreak of a drug-resistant strain of bacteria (Enterococci faecium), hospital officials became concerned that their alcohol-based handrub (ABHR) hand hygiene program was not sufficient to prevent spreading this bacteria. The officials solicited 20 volunteers to assess the effectiveness of ABHR. The volunteers' hands were contaminated with E. faecium. After gathering baseline data on the amount of bacteria present they performed the recommended hand hygiene according to the World Health Organization protocol. The amount of bacteria present was then assessed again. Summary measures and boxplots of the baseline sample, the postABHR sample, and the reduction in the amount of bacteria $\left( \log _ { 10 } ( \right.$ bacteria $\left. / \mathrm { mL } ) \right)$ are presented below. Do these data provide sufficient evidence that the ABHR is effective against the $E$ . faecium?

When analyzing paired data the population distribution of differences, not the separate population distributions, should be credibly normal.

Inferences about the difference between two means fall into two categories: the samples are independent, or the samples are paired. a) What considerations would lead you to use the techniques for independent samples rather than those for paired samples? You may use examples to illustrate your ideas, but examples alone are not sufficient. b) How do the analyses of independent samples and paired samples differ? In your response, consider the hypotheses, methods, assumptions, and calculations.

The Amazon River basin includes the largest remaining connected area of tropical forest in the world, and ecologists regularly monitor its health. One aspect of its health is the density of different indigenous species. The Amazon basin includes both areas that are periodically flooded, and areas that are not periodically flooded. If the densities of species differ in the flooded and unflooded regions, researchers will want to take stratified samples in their studies. Stratified sampling would be more expensive, since boats must be rented to get to any randomly selected flooded sites. The data below are measures of density (individuals per square kilometer) of primates in random samples of flooded and unflooded areas in the Amazon basin. Primate density Unflooded 123 137 106 77 103 79 195 227 182 Flooded 201 126 185 355 344 a) Do these data provide convincing evidence of a difference in the primate density in flooded and unflooded forests in Amazonia? Provide appropriate statistical justification for your conclusion. b) Based on your results in part (a), would you recommend using a stratified sample? Justify your recommendation with a statistical argument.

Male and female Downy Woodpeckers (Picoides pubescens) drill in different areas of trees. One theory about why this separation occurs is that there are physical characteristics of males and females that lead them to choose different foraging locations. One possibility is the bill length of the males and females; longer bills may let one gender or the other drill deeper into a tree. The data in the table at right are the bill lengths of 12 male and 12 female randomly selected Downy Woodpeckers caught and released in a banding survey. The investigators would like to know whether these data provide evidence that the male and female Downy Woodpeckers differ in mean bill length. An initial analysis of the data established the plausibility that the distributions of bill lengths are approximately normal. Downy Woodpecker Bill Lengths (cm) Male Female 2.01 1.78 1.84 1.76 1.86 1.74 1.91 1.82 1.75 1.87 1.79 1.84 1.88 1.82 2.05 1.87 1.85 1.93 1.90 1.76 1.94 1.96 1.86 1.86 a) Construct a $95 \%$ confidence interval for the difference in mean bill length for males and for females. b) Do the data indicate that the mean bill length differs for males and females? Provide an appropriate statistical justification using your response in part (a).

Two samples are said to be independent when the selection of the individuals in one sample has no bearing on the selection of those in the other sample.