Mathematical Modeling in Social Science>Comparing Theoretical and Empirical Probability(TEKS.M.9-12.8.B) Practice Test

A Texas high school baseball coach says a pitcher's mix should be 50% fastballs, 30% curves, and 20% changeups. In one game, 200 pitches were recorded: 88 fastballs, 70 curves, and 42 changeups. At $\alpha=0.05$, which conclusion does a $\chi^2$ goodness-of-fit test support?

At a Texas high school basketball practice, a player shoots two free throws per trip. Assume independent shots with make probability $p=0.75$. Over 160 trips, the coach recorded: 0 made: 16 trips; 1 made: 72 trips; 2 made: 72 trips. Construct the theoretical distribution $X \sim $\text{Bin}$(2,0.75)$, compute the expected counts for each outcome over 160 trips, and identify which outcome shows the largest absolute discrepancy $|$\text{observed}$ - $\text{expected}$|$.