OCR MEI S1 (Statistics 1) 2014 June

The ages, \(x\) years, of the senior members of a running club are summarised in the table below.

Age (\(x\))	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

1. Draw a cumulative frequency diagram to illustrate the data. [5]
2. Use your diagram to estimate the median and interquartile range of the data. [3]

Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities 0.2, 0.5 and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes. [3]
2. Find the probability that a randomly selected candidate is accepted. [2]
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted. [3]

Each weekday, Marta travels to school by bus. Sometimes she arrives late. • \(L\) is the event that Marta arrives late. • \(R\) is the event that it is raining. You are given that \(\mathrm{P}(L) = 0.15\), \(\mathrm{P}(R) = 0.22\) and \(\mathrm{P}(L \mid R) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent. [1]
2. Find \(\mathrm{P}(L \cap R)\). [2]
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]

There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.

1. Find the probability that all 4 members of the team will be girls. [3]
2. Find the probability that the team will contain at least one girl and at least one boy. [3]

The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$

1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]

The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

1. Draw a histogram to illustrate these data. [5]
2. Calculate estimates of the mean and standard deviation of \(w\). [4]
3. Use your answers to part (ii) to investigate whether there are any outliers. [3]

The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$

1. Calculate the mean and standard deviation of \(x\). [3]
2. Compare the central tendency and variation of the weights of varieties A and B. [2]

It is known that on average 85% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.

1. 1. Find the probability that exactly 12 germinate. [3]
   2. Find the probability that fewer than 12 germinate. [2]

The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the 1% significance level to investigate whether he is correct.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
3. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35, complete the test. [3]
4. If \(n\) is small, there is no point in carrying out the test at the 1% significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer. [3]