An examination taken by a large number of students is marked out of a total score of $100$. It is fo

Question

An examination taken by a large number of students is marked out of a total score of $100$. It is found that the mean is $73$ marks and that the standard deviation is $15$ marks. Give a reason why the normal distribution is not a good model for the distribution of marks for the examination. [1] The marks for a random sample of $50$ students is recorded. Find the probability that the mean mark of this sample lies between $70$ and $75$. [2] The interquartile range of a distribution is the difference between the upper and lower quartile values for the distribution. The lower quartile value, $l$, of a distribution $X$, is such that $\mathrm{P}\left( X The marks of another examination is known to follow a normal distribution. If a student who scores $51$ marks is at the $80$th percentile, and the interquartile range is found to be $10.8$ marks, find the mean mark and the standard deviation of the marks scored by students who took the examination. [5] In a third examination, the marks scored by students are normally distributed with a mean of $52$ marks and a standard deviation of $13$ marks. If $50$ is the passing mark and $289$ students are expected to pass, how many candidates are there? [2] CFind the smallest integer value of $m$ such that more than $90\%$ of the candidates will score within $m$ marks of the mean. [3]

Accepted Answer

(a)(i) A normal model assigns probability $0.0359$ to scores above $100$, which are impossible. (ii) $0.748$ (b) $\mu=44.3,\ \sigma=8.01$ (c)(i) $515$ candidates (ii) $m=22$

2017 ACJC P2 Q10

2017 ACJC P2 Q10

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