SPS SPS FM Statistics (SPS FM Statistics) 2026 January

1. At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows. A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the \(1 \%\) significance level.
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2. At a toy factory, wooden blocks of approximate heights \(20 \mathrm {~mm} , 30 \mathrm {~mm}\) and 50 mm are made in red, yellow and green respectively. The heights of the blocks in mm are modelled by independent random variables which are Normally distributed with means and standard deviations as shown in the table. In parts (a), (b) and (c), the blocks are selected randomly and independently of one another.

1. Find the probability that the height of a red block is less than 19 mm .
2. A tower is made of 15 blocks stacked on top of each other consisting of 5 red blocks, 5 yellow blocks and 5 green blocks. Determine the probability that the tower is at least 495 mm high.
3. Determine the probability that a tower made of 3 red blocks will be at least 1 mm higher than a tower made of 2 yellow blocks.
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3. A student is investigating the relationship between different electricity generation methods and cost of electricity in a particular country. The student first checks whether there is any correlation between the cost per unit of electricity, \(x\) euros, and the amount of electricity being generated by wind, \(y \mathrm { GW }\). The data from 30 observations are summarised as follows.
\(n = 30 \quad \sum x = 2.219 \quad \sum y = 357.7 \quad \sum x ^ { 2 } = 0.2368 \quad \sum y ^ { 2 } = 4648 \quad \sum x y = 25.01\)

1. In this question you must show detailed reasoning. Determine the product moment correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) level to investigate whether there is any correlation between the cost per unit of electricity and the amount of electricity generated by wind.
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4. The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   (a) \(\mathrm { P } ( X + Y ) > 40\),
   (b) \(\operatorname { Var } ( 2 X - Y )\).
2. State a necessary assumption for your calculations in part (i) to be valid.
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5. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 3 ^ { 2 } \right)\). A random sample of 9 observations of \(X\) produced the following values. $$\begin{array} { l l l l l l l l l } 6 & 2 & 3 & 6 & 8 & 11 & 12 & 5 & 10 \end{array}$$

1. Find a \(90 \%\) confidence interval for \(\mu\).
2. Explain what is meant by a \(90 \%\) confidence interval in this context.
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6. A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(x\) minutes, by an exponential distribution with probability density function (PDF) $$\mathrm { f } ( x ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda x } & x \geqslant 0
0 & x < 0 \end{cases}$$

1. In this question you must show detailed reasoning. The mean waiting time is found to be 5.0 minutes. Show that \(\lambda = 0.2\).
   ii) Use the model to calculate the probability that a customer has to wait longer than 20 minutes for a response. In practice it is found that no customer waits for more than 15 minutes for a response. The statistician constructs an improved model to incorporate this fact.
   iii) Sketch the following on the same axis.
   (a) the PDF of the model using the exponential distribution,
   (b) a possible PDF for the improved model.
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7. A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.
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8. Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.

1. How many different arrangements of the letters are possible?
2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
3. Find the probability that at least one of these 2 cards has D printed on it.
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9 A continuous random variable \(X\) has probability density function given by the following function, - where \(a\) is a constant.
\(\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise. } \end{array} \right\}\)
The expected value of \(X\) is 4 .

1. Show that \(a = 6\). Five independent observations of \(X\) are obtained, and the largest of them is denoted by \(M\).
2. Find the cumulative distribution function of \(M\).
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