Question: (a) A new university teacher of a large first year class is uncertain as to

how to distribute grades. She decides to assume that the scores

will be Normally distributed (with mean \(\mu\) and standard deviation \(\sigma)\). Her first scheme is to distribute grades as follows: a grade of \(A\) is awarded to those students whose score exceeds \(\mu+\sigma, a B\) to those whose score falls between \(\mu\) and \(\mu +\sigma \), a C to scores

between \(\mu-\sigma\) and a D if a score falls between \(\mu-2 \sigma\) and \(\mu-\sigma\)

and an \(\mathrm{F}\) for scores below \(\mu-2 \sigma\)

1. What proportion of students get A under this scheme?
2. What cut-off is required for A grades if she wants to limit the

proportion of A grades to \(5 \%\) ?

(b) Suppose that the scores are Normally distributed with mean 50 and

standard deviation 15

1. What percentage of students fail, if the pass mark is 40?
2. If a random sample of three scripts is selected by the external examiner for detailed scrutiny, what is the probability that the average score for these scripts will be less than 40?
3. Explain why it would be surprising if the sum of the three

scores exceeded 250.

(c) Normal (probability) plots are used to assess data Normality. Explain the logic underlying these plots. Draw a rough sketch to illustrate what a Normal plot might look like if the data came from a skewed distribution. The Minitab implementation of the Normal plot provides an Anderson—Darling test, together with a p-value. Explain the interpretation of p-values of 0.24 and 0.024 in this context.

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