OCR MEI Further Statistics A AS (Further Statistics A AS) Specimen

1 The number of failures of a machine each week at a factory is modelled by a Poisson distribution with mean 0.45.

1. Write down the variance of the distribution.
2. Find the probability that there are exactly 2 failures in a week.
3. State a distribution which can be used to model the number of failures in a period of 4 weeks.
4. Find the probability that there are at least 2 failures in a period of 4 weeks.

2 The discrete random variable \(Y\) is uniformly distributed over the values \(\{ 12,13 , \ldots , 20 \}\).

1. Write down \(\mathrm { P } ( Y < 15 )\).
2. Two independent observations of \(Y\) are taken. Find the probability that one of these values is less than 15 and the other is greater than 15 .
3. Find \(\mathrm { P } ( Y > \mathrm { E } ( Y ) )\).

3 In this question you must show detailed reasoning. A student is investigating what people think about organic food. She wishes to see if there is any difference between the opinions of females and males. She takes a random sample of 100 people and asks each of them if they think that organic food is better for their health than non-organic food. She will use the data to conduct a hypothesis test. The table below shows the opinions of these 100 people.

1. Explain why the student should use a random sample.
2. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a person's sex and their opinion on organic food. Show your calculations.

4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\).
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find
   • the mean of the total of the three scores.
   • the variance of the total of the three scores.

5 In a recent report, it was stated that \(40 \%\) of working people have a degree. For the whole of this question, you should assume that this is true. A researcher wishes to interview a working person who has a degree. He asks working people at random whether they have a degree and counts the number of people he has to ask until he finds one with a degree.

1. Find the probability that he has to ask 5 people.
2. Find the mean number of people the researcher has to ask. Subsequently, the researcher decides to take a random sample from the population of working people.
3. A random sample of 5 working people is chosen. What is the probability that at least one of them has a degree?
4. How large a random sample of working people would the researcher need to take to ensure that the probability that at least one person has a degree is 0.99 or more?

6 A motorist decides to check the fuel consumption, \(y\) miles per gallon, of her car at particular speeds, \(x \mathrm { mph }\), on flat roads. She carries out the check on a suitable stretch of motorway. Fig. 6 shows her results. \begin{figure}[h] \end{figure}

1. Explain why it would not be appropriate to carry out a hypothesis test for correlation based on the product moment correlation coefficient.
2. (A) One of the results is an outlier. Circle the outlier on the copy of Fig. 6 in the Printed Answer Booklet.
   (B) Suggest one possible reason for the outlier in part (ii) (A) not being used in any analysis. The motorist decides to remove this item of data from any analysis. The table below shows part of a spreadsheet that was used to analyse the 14 remaining data items (with the outlier removed). Some rows of the spreadsheet have been deliberately omitted.

   	Data item	\(x\)	\(y\)	\(x ^ { 2 }\)	\(y ^ { 2 }\)	\(x y\)
   	1	50	53.6	2500	2872.96	2680
   	2	50	53.3	2500	2840.89	2665
   	13	70	44.8	4900	2007.04	3136
   	14	70	44.2	4900	1953.64	3094
   Sum		840	686	51150	33779.7	40812

3. Calculate the equation of the regression line of \(y\) on \(x\).
4. Use the equation of the regression line to predict the fuel consumption of the car at
   (A) 58 mph ,
   (B) 30 mph .
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