Final answer:
The probabilities and percentages related to the mathematics program, law school eligibility, and John Abbott College awards based on R-scores are calculated using the standard normal distribution, Z-scores, and assuming a normal distribution with a given mean and standard deviation.
Step-by-step explanation:
Calculating Probabilities for Normal Distribution
To calculate the probability that a random student is not eligible for a mathematics program requiring an R-score of at least 28, we would use the standard normal distribution. Since we know the mean (μ) is 25 and the standard deviation (σ) is unspecified, you would typically use a Z-table or statistical software to find P(X < 28). To do so, you would find the Z-score for an R-score of 28 using the formula Z = (X - μ) / σ, where X is the R-score.
Similarly, for law school eligibility, you would calculate the percentage of students with R-scores greater than or equal to the required threshold, again assuming you have the necessary standard deviation. Subtract this probability from 1 to find the percentage that is not eligible.
For part (c), finding the R-score for students in the top 7%, you would use a Z-score table or technology to find the Z-score that corresponds to the cumulative area of 0.93 (since we want the top 7%, the total area under the curve to the left of our Z-score would be 1 - 0.07). The R-score threshold for awards would be the Z-score multiplied by the standard deviation, plus the mean.