Justifying CLT for hypothesis testing

A question is this sub-type if and only if it asks whether CLT was necessary (or requires stating assumptions) when performing a hypothesis test about a population mean with unknown population distribution.

2 questions · Standard +0.6

5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean
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CAIE S2 2013 June Q7
14 marks Standard +0.8
7 In the past the weekly profit at a store had mean \(\\) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(\\) 35400$.
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
  2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
  3. State the probability of a Type I error.
  4. Given that the population mean weekly profit is now \(\\) 36500$, calculate the probability of a Type II error.
OCR S2 2013 January Q6
10 marks Standard +0.3
6 Gordon is a cricketer. Over a long period he knows that his population mean score, in number of runs per innings, is 28 , and the population standard deviation is 12 . In a new season he adopts a different batting style and he finds that in 30 innings using this style his mean score is 28.98 .
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether his population mean score has increased.
  2. Explain whether it was necessary to use the Central Limit Theorem in part (i).