OCR MEI S1 (Statistics 1) 2010 January

1 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(50,60 , \ldots\).
2. Write down the median and midrange of the data.
3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.

2 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).

1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
   (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).

   \(r\)	10	15	20	25	30
   \(\mathrm { P } ( X = r )\)	0.1	0.4	0.1	0.2	0.2

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

3 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.

• \(G\) is the event that this person goes to the gym.
• \(R\) is the event that this person goes running.

You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
2. Determine whether the events \(G\) and \(R\) are independent.
3. Find \(\mathrm { P } ( R \mid G )\).

4 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.

1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.

6 Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if

1. no student may win more than 1 prize,
2. no student may win all 3 prizes. Section B (36 marks)

7 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{2f39c509-5429-4193-9526-15fb45b18a38-4_837_1651_466_246}

1. Calculate the number of pears which are between 90 and 100 mm long.
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
3. Calculate an estimate of the standard deviation.
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
5. Name the type of skewness of the distribution.
6. Illustrate the data using a cumulative frequency diagram.

8 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

1. Three days are chosen at random. Find the probability that the pollution level is
   (A) low on all 3 days,
   (B) low on at least one day,
   (C) low on one day, medium on another day, and high on the other day.
2. Ten days are chosen at random. Find the probability that
   (A) there are no days when the pollution level is high,
   (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.