A New Type of Band Has Been Developed for Children

Essay

A new type of band has been developed for children who have to wear braces. The new
bands are designed to be more comfortable, look better, and provide more rapid progress
in realigning teeth. An experiment was conducted to compare the mean wearing time
necessary to correct a specific type of misalignment between the old braces and the new
bands. One hundred children were randomly assigned, 50 to each group. A summary of
the data is shown in the table. $$\begin{array} { c | c c } & \text { Old Braces } & \text { New Bands } \\\hline \bar { x } & 410 \text { days } & 380 \text { days } \\s & 41 \text { days } & 64 \text { days }\end{array}$$ How many patients would need to be sampled to estimate the difference in means to
within 31 days with probability 99%?