OCR MEI Further Statistics A AS (Further Statistics A AS) 2018 June

1 Over a period of time, radioactive substances decay into other substances. During this decay a Geiger counter can be used to detect the number of radioactive particles that the substance emits. A certain radioactive source is decaying at a constant average rate of 6.1 particles per 10 seconds. The particles are emitted randomly and independently of each other.

1. State a distribution which can be used to model the number of particles emitted by the source in a 10-second period.
2. State the variance of this distribution.
3. Find the probability that at least 6 particles are detected in a period of 10 seconds.
4. Find the probability that at least 36 particles are detected in a period of 60 seconds.
5. Another radioactive source emits particles randomly and independently at a constant average rate of 1.7 particles per 5 seconds. Find the probability that at least 10 but no more than 15 particles are detected altogether from the two sources in a period of 10 seconds.

2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable \(X\) represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of \(X\) is given in the following table.

1. Explain why \(\mathrm { P } ( X = 4 )\) must be 0 .
2. Explain how the value \(\frac { 1 } { 120 }\) for \(\mathrm { P } ( X = 5 )\) is calculated.
3. Draw a graph to illustrate the distribution.
4. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
   • There are 12 competitors in the quiz. Assuming that they all guess randomly, find the probability that at least one of them gets all five matches correct.

3 Samples of water are taken from 10 randomly chosen wells in an area of a country. A researcher is investigating whether there is any relationship between the levels of dissolved oxygen, \(x\), and the amounts of radium, \(y\), in the water from the wells. Both quantities are measured in suitable units. The table and the scatter diagram in Fig. 3 show the values of \(x\) and \(y\) for the ten wells. \begin{figure}[h] \end{figure}

1. Explain why it may not be appropriate to carry out a hypothesis test based on the product moment correlation coefficient.
2. Calculate Spearman's rank correlation coefficient for these data.
3. Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the 1\% significance level to investigate whether there is any association between \(x\) and \(y\).
4. Explain the meaning of the term 'significance level' in the context of the test carried out in part (iii).

4 The probability that an expert darts player hits the bullseye on any throw is 0.12 , independently of any other throw. The player throws darts at the bullseye until she hits it.

1. Find the probability that the player has to throw exactly six darts.
2. Find the probability that the player has to throw more than six darts.
3. (A) Find the mean number of darts that the player has to throw.
   (B) Find the variance of the number of darts that the player has to throw. The player continues to throw more darts at the bullseye after she has hit it for the first time.
4. Find the probability that the player hits the bullseye at least twice in the first ten throws.
5. Find the probability that the player hits the bullseye for the second time on the tenth throw.

5 A random sample of workers for a large company were asked whether they are smokers, ex-smokers or have never smoked. The responses were classified by the type of worker: Managerial, Production line or Administrative. Fig. 5 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h] \end{table}

1. (A) State the sample size.
   (B) State the null and alternative hypotheses for a test to investigate whether there is any association between type of worker and smoking status.
2. Showing your calculations, find the missing values in each of the following cells.
   • C 10
   • C 15
   • B 16
   • Complete the hypothesis test at the \(10 \%\) level of significance.
   • Discuss briefly what the data suggest about smoking status for different types of workers. You should make a comment for each type of worker.

6 A researcher is investigating various bodily characteristics of frogs of various species. She collects data on length, \(x \mathrm {~mm}\), and head width, \(y \mathrm {~mm}\), of a random sample of 14 frogs of a particular species. A scatter diagram of the data is shown in Fig. 6, together with the equation of the regression line of \(y\) on \(x\) and also the value of \(r ^ { 2 }\). \begin{figure}[h] \end{figure}

1. (A) Use the equation of the regression line to estimate the mean head width for frogs of each of the following lengths.
   • 45 mm
   • 60 mm
     (B) Comment briefly on each of the estimates in part (i)(A).
   • Explain how the mean length of frogs with head width 16 mm should be estimated.
   • Calculate the value of the product moment correlation coefficient.
   • In the light of the information in the scatter diagram, comment on the goodness of fit of the regression line.