The Last Scholarship

Look at the three calculations you just did in Task 1c. Write a single algebraic rule that takes any score \(x\), mean \(\mu\), and standard deviation \(\sigma\), and produces "the number of standard deviations from the mean." Test your rule against your three Phase 1 answers to check it agrees.

Use your rule on each of these. Watch what happens when the score is below the mean, and when it's exactly at the mean.

Here is a tiny dataset: 4, 6, 7, 9. Find its mean and standard deviation. Then apply your rule from Task 2a to every single value in it, turning it into four new numbers.

Now find the mean and standard deviation of your four new numbers. What do you notice? Do you think this would happen for any dataset, or just this one?

If two finalists (from two completely different exams) had exactly the same z-score, would they be equally rare within their own exam's population? Discuss as a group and write your prediction, yes or no, with a reason, on your board.

Use your GDC to find \(P(X > 132)\) directly, where \(X\) is Amara's exam, \(X \sim \text{N}(100, 15^2)\). Then find \(P(Z > z)\) using the standard normal distribution, with the z-score you found for Amara in Phase 1. Compare the two answers.

Sketch both curves on your board (Amara's exam, and the standard normal), shade the matching region on each, and compare their shapes.

Now find \(P(Z > z)\) for all three finalists: the probability of scoring that well or better within their own exam. Rank the three finalists from most exceptional to least exceptional. Does this match your Phase 1 gut vote?

As a committee, decide who should get the scholarship, and justify it using these probabilities, not the raw scores.

Sketch the standard normal curve once and mark all three finalists' cutoffs on it, shading the smallest region.

Before you can find \(\sigma\), you need the z-value that matches "only 2% scored above 145," that is, the value \(z\) such that \(P(Z > z) = 0.02\). Use your GDC's inverse normal function to find it.

Sketch the standard normal curve, shade the right-tail region worth 0.02 of the area, and mark the boundary z on it.

Using \(z = \dfrac{x-\mu}{\sigma}\) with \(x = 145\), \(\mu = 110\), and the \(z\) you just found, rearrange and solve for \(\sigma\). Keep full calculator precision for \(z\); don't round it before dividing.

Now that Exam D has a working \(\sigma\), find Jordan's z-score using her actual score, \(x = 150\). Compare her z-score (and probability) to Amara, Diego, and Priya's from Phase 3. Does the scholarship decision change?

Add Jordan's cutoff to your Phase 3 sketch, so all four finalists are marked on the same curve.