Male and Female Downy Woodpeckers (Picoides Pubescens) Drill in Different

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)
$\begin{array}{|c|c|}\hline \text { Male } & \text { Female } \\\hline 2.01 & 1.78 \\1.84 & 1.76 \\1.86 & 1.74 \\1.91 & 1.82 \\1.75 & 1.87 \\1.79 & 1.84 \\1.88 & 1.82 \\2.05 & 1.87 \\1.85 & 1.93 \\1.90 & 1.76 \\1.94 & 1.96 \\1.86 & 1.86 \\\hline\end{array}$


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).