OCR MEI S1 (Statistics 1) 2010 January

A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$73 \quad 67 \quad 75 \quad 64 \quad 52 \quad 63 \quad 75 \quad 81 \quad 77 \quad 72 \quad 68 \quad 74 \quad 79 \quad 72 \quad 71$$

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of 50, 60, ... . [4]
2. Write down the median and midrange of the data. [2]
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 £5 notes, two £10 notes and one £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 \(£X\).

1. 1. Show that P(X = 10) = 0.1. [1]
   2. Show that P(X = 30) = 0.2. [2]
   The table shows the probability distribution of X.

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

2. Find E(X) and Var(X). [5]

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 P(G) = 0.24, P(R) = 0.13 and P(G ∩ 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. [3]
2. Determine whether the events \(G\) and \(R\) are independent. [2]
3. Find P(R | G). [3]

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. [3]
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct. [2]

My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.

1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. [1]

In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.

1. If I now guess my number, find the probability that I guess it correctly. [2]

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]
2. no student may win all 3 prizes. [2]

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{figure_7}

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

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
   1. low on all 3 days, [2]
   2. low on at least one day, [2]
   3. low on one day, medium on another day, and high on the other day. [3]
2. Ten days are chosen at random. Find the probability that
   1. there are no days when the pollution level is high, [2]
   2. there is exactly one day when the pollution level is high. [3]

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.

1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]