Problem
The "Freshman $15^{\prime \prime}$ refers to the belief that college students gain $15 \mathrm{lb}$ (or $6.8 \mathrm{~kg}$ ) during their freshman year. Listed in the accompanying table are weights $(\mathrm{kg})$ of randomly selected male college freshmen. The weights were measured in September and later in April. Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. Complete parts (a) through (c). $\begin{array}{llllllllll}\text { September } & 53 & 97 & 68 & 63 & 42 & 57 & 56 & 71 & 70 \\ \text { April } & 56 & 86 & 68 & 66 & 49 & 58 & 58 & 77 & 68\end{array}$ hypotheses for the hypothesis test? \[ \begin{array}{l} H_{0}: \mu_{d}=0 \mathrm{~kg} \\ H_{1}: \mu_{d}>0 \mathrm{~kg} \end{array} \] (Type integers or decimals. Do not round.) Identify the test statistic. $\mathrm{t}=\square$ (Round to two decimal places as needed.)
Solution
Step 1 ：First, calculate the differences between the weights in April and September for each student. The differences are [3, -11, 0, 3, 7, 1, 2, 6, -2].
Step 2 ：Next, calculate the mean and standard deviation of these differences. The mean difference is 1.0 and the standard deviation is approximately 5.29.
Step 3 ：Then, identify the number of pairs, which is 9 in this case.
Step 4 ：Set the hypothesized mean difference to 0.
Step 5 ：Finally, substitute these values into the formula to find the test statistic. The test statistic for the hypothesis test is approximately \(0.57\).
