OCR MEI S1 — Question 3 18 marks

Exam BoardOCR MEI
ModuleS1 (Statistics 1)
Marks18
PaperDownload PDF ↗
TopicHypothesis test of binomial distributions
TypeOne-tailed test critical region
DifficultyStandard +0.3 This is a straightforward application of binomial probability and hypothesis testing with standard procedures. Part (i) involves routine binomial calculations, while parts (ii)-(iii) require setting up a one-tailed test and finding a critical region—all textbook exercises with no novel problem-solving required. Slightly above average difficulty due to the multi-step nature and hypothesis testing component, but remains a standard S1 question.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail
3 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.
  1. 15 children are selected at random.
    (A) Find the probability that exactly 3 of them eat at least five a day.
    (B) Find the probability that at least 3 of them eat at least five a day.
    (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.
3 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, $20 \%$ of pupils eat at least five a day.
\begin{enumerate}[label=(\roman*)]
\item 15 children are selected at random.\\
(A) Find the probability that exactly 3 of them eat at least five a day.\\
(B) Find the probability that at least 3 of them eat at least five a day.\\
(C) Find the expected number who eat at least five a day.

A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
\item (A) Write down suitable null and alternative hypotheses for the test.\\
(B) Give a reason for your choice of the alternative hypothesis.
\item Find the critical region for the test at the $10 \%$ significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S1  Q3 [18]}}
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Q1 5 Q2 19 Q3 18