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

1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing. Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them. \begin{table}[h] \end{table}

1. (A) Write down the numbers in columns H , I and J for game 20 .
   (B) Use the spreadsheet to estimate the probability of a player winning a game.
2. State how the estimate of probability in (i) (B) could be improved.
3. Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability. All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
4. If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.

2 The cumulative distribution function of the continuous random variable, \(Y\), is given below. $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { y ^ { 3 } - y ^ { 2 } } { 4 } & 1 \leq y \leq 2
1 & y > 2 \end{array} \right.$$

1. Find \(\mathrm { P } ( Y \leq 1.5 )\)
2. Verify that the median of \(Y\) lies between 1.6 and 1.7.
3. Find the probability density function of \(Y\).

3 At a factory, flour is packed into bags. A model for the mass in grams of flour packed into each bag is \(1500 + X\), where \(X\) is a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x ( 6 - x ) & 0 \leq x \leq 6
0 & \text { elsewhere, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 36 }\).
2. Find the probability that a randomly selected bag of flour contains 1505 grams of flour or more.
3. Find
   • the mean of \(X\),
   • the standard deviation of \(X\).

4 An online encyclopedia claims that the average mass of an adult European hedgehog is 720 g . In an investigation to check this average figure, the masses in grams of twelve randomly chosen adult European hedgehogs are measured and shown below.

1. What assumption is required to carry out a Wilcoxon test in this situation?
2. Given that this assumption is met, carry out a 2 -tail Wilcoxon test at the \(5 \%\) level to test whether the median mass is 720 g . You should state your hypotheses and complete the table of calculations in the Printed Answer Booklet.

5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows. The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1. \begin{figure}[h] \end{figure}

1. Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data. The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726} \captionsetup{labelformat=empty} \caption{Fig 5.2}
   \end{figure}
2. State the confidence interval which the software gives, in the form \(a < \mu < b\).
3. (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
   (B) State whether the assumption in part (iii) (A) seems reasonable.
4. Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
5. In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
   (A) Explain what a standard error is.
   (B) Show how the standard error was calculated in this case.
6. Suggest a way in which the researcher could produce a narrower confidence interval.

6 The table below shows the mean and variance of the test scores of a random samples of 70 girls who are starting an A level Mathematics course.

1. Showing your working, find a \(95 \%\) confidence interval for the population mean.
2. Explain why you can construct the interval in part (i) despite no information about the distribution of the parent population being given.
3. The same random sample of girls repeats the test. The mean improvement in score is 0.9 . The \(95 \%\) confidence interval for the improvement is \([ - 1.5,3.3 ]\). What is the sample variance for the improvement in score?

7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.

• By bus: Normally distributed with mean 23 and standard deviation 6
• By bicycle: Normally distributed with mean 21 and standard deviation 2

You should assume that all journey times are independent.

1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
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