### Find and compare two standard scores and scores in two normally distributed examinations

Question Sample Titled 'Find and compare two standard scores and scores in two normally distributed examinations'

The table below shows the means and the standard deviations of the scores of a large group of students in a Mathematics examination and a Biology examination:


ExaminationMeanStandard deviation
Mathematics${58}$ marks${14}$ marks
Biology${68}$ marks${15}$ marks

The standard score of Ethan in the Mathematics examination is $-{1.5}$ .

 (a) Find the score of Ethan in the Mathematics examination. (2 marks) (b) Assume that the scores in each of the above examinations are normally distributed. Ethan gets ${56}$ marks in the Biology examination. He claims that relative to other students, he performs better in the Biology examination than in the Mathematics examination. Is the claim correct? Explain your answer.  (2 marks)

 (a) Let ${x}$ marks be the score of Ethan in the Mathematics examination. $\dfrac{{{x}-{58}}}{{14}}$ $=-{1.5}$ 1M ${x}$ $={58}+{\left(-{1.5}\right)}{\left({14}\right)}$ ${x}$ $={37}$ 1A Thus, the score of Ethan in the Mathematics examination is ${37}$ marks. (b) The standard score of Ethan in the Biology examination $=\dfrac{{{56}-{68}}}{{15}}$ 1M $=-{0.8}$ $\lt-{1.5}$ Relative to other students, Ethan performs better in the Biology examination than in the Mathematics examination. Thus, the claim is correct. 1A f.t. Caution: Comparing the two absolute difference of Ethan's marks to the means should be awarded zero marks.

 (b) Note that, relative to other students, if Ethan performs equally in both examination, the standard score of both examination should be the same. Let ${y}$ be his score in Biology examination so that Ethan performs equally in both examination. $\dfrac{{{y}-{68}}}{{15}}$ $=-{1.5}$ 1M ${y}$ $={45.5}$ He gets ${56}$ marks in Biology examination, which is higher than ${45.5}$ marks. Therefore, relative to other students, Ethan performs better in the Biology examination than in the Mathematics examination. Thus, the claim is correct. 1A f.t.

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