7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by
\(n = 50\),
\(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\),
\(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
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- Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
- Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
- Explain whether the assumption that times are normally distributed is necessary in answering
(a) part (i),
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(b) part (ii). [2]