SPS SPS FM Statistics (SPS FM Statistics) 2022 February

1. The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   On average \(65 \%\) of eggs laid result in a young bird successfully leaving the nest.
2. 1. Find the mean number of young birds that successfully leave the nest.
   2. Find the standard deviation of the number of young birds that successfully leave the nest.
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2. A shopper estimates the cost, \(\pounds X\) per item, of each of 12 items in a supermarket. The shopper's estimates are compared with the actual cost, \(\pounds Y\) per item, of each item. The results are summarised as follows.
\(n = 12\)
\(\sum x ^ { 2 } = 28127\)
\(\sum x = 399\)
\(\Sigma y ^ { 2 } = 116509.0212\)
\(\Sigma y = 623.88\)
\(\sum x y = 45006.01\) Test at the \(1 \%\) significance level whether the shopper's estimates are positively correlated with the actual cost of the items.
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3. A football player is practising taking penalties. On each attempt the player has a \(70 \%\) chance of scoring a goal. The random variable \(X\) represents the number of attempts that it takes for the player to score a goal.

1. Determine \(\mathrm { P } ( X = 4 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Determine the probability that the player needs exactly 4 attempts to score 2 goals.
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1. (a) Using the scatter diagram below, explain what is meant by least squares in the context of a regression line of \(y\) on \(x\).
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   (b) A set of bivariate data \(( t , u )\) is summarised as follows.

$$\begin{array} { l l l } n = 5 & \sum t = 35 & \sum u = 54
\sum t ^ { 2 } = 285 & \sum u ^ { 2 } = 758 & \sum t u = 460 \end{array}$$

1. Calculate the equation of the regression line of \(u\) on \(t\).
2. The variables \(t\) and \(u\) are now scaled using the following scaling. $$v = 2 t , w = u + 4$$ Find the equation of the regression line of \(w\) on \(v\), giving your equation in the form $$w = \mathrm { f } ( v ) .$$ [BLANK PAGE]

5. Charlie carried out a survey on the main type of investment people have. The contingency table below shows the results of a survey of a random sample of people.

1. Find an expression, in terms of \(a , b , c\) and \(d\), for the difference between the observed and the expected value \(( O - E )\) for the group whose main type of investment is Bonds and are aged \(45 - 75\)
   Express your answer as a single fraction in its simplest form. Given that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62\) for this information,
2. test, at the \(5 \%\) level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
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6. The 20 members of a club consist of 10 Town members and 10 Country members.

1. All 20 members are arranged randomly in a straight line. Determine the probability that the Town members and the Country members alternate.
2. Ten members of the club are chosen at random. Show that the probability that the number of Town members chosen is no more than \(r\), where \(0 \leqslant r \leqslant 10\), is given by
   \(\frac { 1 } { N } \sum _ { i = 0 } ^ { r } \left( { } ^ { 10 } C _ { i } \right) ^ { 2 }\)
   where \(N\) is an integer to be determined.
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7. (a) A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
(b) The random variable \(X\) represents the number of particles emitted by a substance in a fixed time interval \(t\) minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for \(X\) if the value of \(t\) is very large.
(c) The random variable \(Y\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\begin{aligned} & \mathrm { P } ( Y = r ) = \mathrm { P } ( Y = r + 1 )
& \mathrm { P } ( Y = r ) = 1.5 \times \mathrm { P } ( Y = r - 1 ) \end{aligned}$$ Determine the following, in either order.

• The value of \(r\)
• The value of \(\lambda\)
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