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.
$$\begin{array} { c | r c } & \text { Old Braces } & \text { New Bands } \\\hline \bar { x } & 410 \text { days } & 380 \text { days } \\s & 41 \text { days } & 65 \text { days }\end{array}$$

How many patients would need to be sampled to estimate the difference in means to within 26 days with probability 99%?