OCR MEI S1 (Statistics 1) 2013 June

1 The weights, \(x\) grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$

1. Calculate the mean and standard deviation of \(x\).
2. The weights, \(y\) grams, of the potatoes after they have been peeled are given by the formula \(y = 0.9 x - 15\). Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.

1. Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
2. Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.

3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average \(10 \%\) of bags are underweight.

1. Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
2. Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
3. A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain \(\pounds 5\) and the other three each contain \(\pounds 1000\). Since the envelopes are identical on the outside, he chooses three of them at random. Let \(\pounds X\) be the total amount of money that he receives in prize money.

1. Show that \(\mathrm { P } ( X = 15 ) = 0.05\). The probability distribution of \(X\) is given in the table below.

   \(r\)	15	1010	2005	3000
   \(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.

1. Explain why the null hypothesis should be \(p = 0.5\), where \(p\) represents the probability that a randomly selected person makes a correct identification.
2. Briefly explain why she uses an alternative hypothesis of \(p > 0.5\).
3. Complete the test at the \(5 \%\) significance level.

6 The birth weights in kilograms of 25 female babies are shown below, in ascending order.

1. Find the median and interquartile range of these data.
2. Draw a box and whisker plot to illustrate the data.
3. Show that there is exactly one outlier. Discuss whether this outlier should be removed from the data. The cumulative frequency curve below illustrates the birth weights of 200 male babies.
   \includegraphics[max width=\textwidth, alt={}, center]{6b886da6-3fb8-4b4c-b572-f4b770ae5a4c-3_929_1569_1450_248}
4. Find the median and interquartile range of the birth weights of the male babies.
5. Compare the weights of the female and male babies.
6. Two of these male babies are chosen at random. Calculate an estimate of the probability that both of these babies weigh more than any of the female babies.

7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.

1. Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
2. Find the probability that
   (A) she hits the bull's-eye with at least one of her first three darts,
   (B) she hits the bull's-eye with exactly one of her first three darts.
3. Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
4. Find the probability that she hits the bull's-eye three times in total.