A university has one scholarship spot left. Three finalists have made it to the final round, each holding the top score from a completely different national exam. Keep this card, you will need this data for all four phases.
| Finalist | Exam | Score (x) | Mean (渭) | SD (蟽) |
|---|---|---|---|---|
| Amara | Exam A | 132 | 100 | 15 |
| Diego | Exam B | 78 | 65 | 8 |
| Priya | Exam C | 610 | 500 | 100 |
Without calculating: which finalist should get the scholarship? Write your pick and reasoning.
Why might comparing 610, 132, and 78 directly be a mistake? Give a real-world comparison with the same problem.
For each finalist, work out how many SDs above or below average they scored, using only their score, mean, and SD. Sketch a bell curve and mark where each finalist sits.
Write a single formula in \(x\), \(\mu\), \(\sigma\) that gives "SDs from the mean." Test it against your Task 1c values.
Apply your formula: (54, 50, 2) 路 (42, 50, 4) 路 (50, 50, 8) 路 (127, 100, 18) 路 (88, 100, 16). Watch the signs.
Standardize the dataset 4, 6, 7, 9 using your rule. Find the mean and SD of your four new values. What do you notice?
Write \(Z=\frac{X-\mu}{\sigma}\) as \(aX+b\). Use \(\text{E}(aX+b)=a\text{E}(X)+b\) and \(\text{Var}(aX+b)=a^2\text{Var}(X)\) to prove \(\text{E}(Z)=0\), \(\text{Var}(Z)=1\) for any \(\mu, \sigma\).
If two finalists from different exams had the same z-score, would they be equally rare? Predict, with a reason.
Use your GDC: find \(P(X>132)\) on \(N(100,15^2)\), then \(P(Z>z)\) on the standard normal. Sketch both curves, shaded, and compare.
Find \(P(Z>z)\) for all three finalists. Rank them. As a committee, decide who wins, and justify it with probability, not raw score.
Find \(z\) such that \(P(Z>z)=0.02\) using inverse normal. Sketch the curve, shaded, with \(z\) marked.
Rearrange \(z=\frac{x-\mu}{\sigma}\) with \(x=145\), \(\mu=110\), and your \(z\), to find \(\sigma\). Keep full precision.
Find Jordan's z-score using \(x=150\). Compare to Amara, Diego, and Priya. Does the scholarship decision change?