A teacher gives her students the same test at the beginning and end of the week. She wants to know if their scores increase over time. The summary statistics that she has collected are below. \( \bar{x}_{d}=4.2 \) \( s_{d}=4.3 \) \( n=35 \) \( H_{o}: \mu_{d}=0 \quad H_{a}: \mu_{d}>0 \) Given the summary statistics and hypotheses above, compute the \( t \) test statistic. Round your final answer to 3 decimal places.

To compute the \( t \) test statistic, we can use the formula: \[ t = \frac{\bar{x}_{d}}{s_{d}/\sqrt{n}} \] Where: - \( \bar{x}_{d} = 4.2 \) (the mean difference between the two tests) - \( s_{d} = 4.3 \) (the standard deviation of the differences) - \( n = 35 \) (the number of students) Now, plug in the values: 1. Calculate \( s_{d}/\sqrt{n} \): \[ s_{d}/\sqrt{n} = 4.3/\sqrt{35} \approx 4.3/5.916 = 0.727 \] 2. Now, calculate the \( t \) statistic: \[ t = \frac{4.2}{0.727} \approx 5.775 \] So, rounding to three decimal places, the \( t \) test statistic is: \[ \boxed{5.775} \]