Edexcel FS2 (Further Statistics 2) 2023 June

1. Baako is investigating the times taken by children to run a 100 m race, \(x\) seconds, and a 500 m race, \(y\) seconds. For a sample of 20 children, Baako obtains the time taken by each child to run each race.

Here are Baako's summary statistics. $$\begin{gathered} \mathrm { S } _ { x x } = 314.55 \quad \mathrm {~S} _ { y y } = 9026 \quad \mathrm {~S} _ { x y } = 1610
\bar { x } = 19.65 \quad \bar { y } = 108 \end{gathered}$$

1. Calculate the product moment correlation coefficient between the times taken to run the 100 m race and the times taken to run the 500 m race.
2. Show that the equation of the regression line of \(y\) on \(x\) can be written as $$y = 5.12 x + 7.42$$ where the gradient and \(y\) intercept are given to 3 significant figures. The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.
3. Find the residual for this child. The table below shows the signs of the residuals for the 20 children in order of finishing time for the 100 m race.

   Sign of residual

   +

   +

   +

   +

   -

   -

   +

   -

   -

   -

   -

   -

   -

   -

   -

   +

   +

   +

   +

   +

4. Explain what the signs of the residuals show about the model's predictions of the 500 m race times for the children who are fastest and slowest over the 100 m race.

1. Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.
A random sample of 80 green apples has a mean weight of 128 grams.

1. Find a 98\% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places. Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples. A random sample of \(n\) red apples has a mean weight of 117 grams.
   The standard deviation of the weights of the red apples is known to be 4 grams.
   A test of Camilo's belief is carried out at the 5\% level of significance.
2. State the null and alternative hypotheses for this test.
3. Find the smallest value of \(n\) for which the null hypothesis will be rejected.
4. Explain the relevance of the Central Limit Theorem in parts (a) and (c).
5. Given that \(n = 85\), state the conclusion of the hypothesis test.

1. Two machines, \(A\) and \(B\), are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.

Samples are taken from each machine and the amount of water, \(x \mathrm { ml }\), dispensed in each bottle is recorded. The table shows the summary statistics for Machine \(A\).

1. Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine \(A\). For Machine \(B\), a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is \(12.7 \mathrm { ml } ^ { 2 }\)
2. Test, at the \(10 \%\) level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.

1. The weights of eggs, \(E\) grams, follow a normal distribution, \(\mathrm { N } \left( 60,3 ^ { 2 } \right)\)

The weights of empty small boxes, \(S\) grams, follow a normal distribution, \(\mathrm { N } \left( 24,1.8 ^ { 2 } \right)\)
The weights of empty large boxes, \(L\) grams, follow a normal distribution, \(\mathrm { N } \left( 40,2.1 ^ { 2 } \right)\)
Small boxes of eggs contain 6 randomly selected eggs.
Large boxes of eggs contain 12 randomly selected eggs.

1. Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.
2. Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.

1. A psychologist claims to have developed a technique to improve a person's memory.

A random sample of 8 people are each given the same list of words to memorise and recall. Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall. The table shows the percentage of words recalled by each person before and after the training.

1. State why a paired \(t\)-test is suitable for these data.
2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this test.

1. The continuous random variable \(X\) has cumulative distribution function given by

$$F ( x ) = \left\{ \begin{array} { c r } 0 & x < 0
k \left( x - a x ^ { 2 } \right) & 0 \leqslant x \leqslant 4
1 & x > 4 \end{array} \right.$$ The values of \(a\) and \(k\) are positive constants such that \(\mathrm { P } ( X < 2 ) = \frac { 2 } { 3 }\)

1. Find the exact value of the median of \(X\)
2. Find the probability density function of \(X\)
3. Hence, deduce the value of the mode of \(X\), giving a reason for your answer.

1. The random variable \(R\) has a continuous uniform distribution over the interval \([ 2,10 ]\)
   1. Write down the probability density function \(\mathrm { f } ( r )\) of \(R\)
   A sphere of radius \(R \mathrm {~cm}\) is formed.
   The surface area of the sphere, \(S \mathrm {~cm} ^ { 2 }\), is given by \(S = 4 \pi R ^ { 2 }\)
2. Show that \(\mathrm { E } ( S ) = \frac { 496 \pi } { 3 }\) The volume of the sphere, \(V \mathrm {~cm} ^ { 3 }\), is given by \(V = \frac { 4 } { 3 } \pi R ^ { 3 }\)
3. Find, using algebraic integration, the expected value of \(V\)

1. A bag contains a large number of marbles of which an unknown proportion, \(p\), is yellow.

Three random samples of size \(n\) are taken, and the number of yellow marbles in each sample, \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\), is recorded. Two estimators \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are proposed to estimate the value of \(p\) $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n }
& \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$

1. Show that \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are both unbiased estimators of \(p\)
2. Find the variance of \(\hat { p } _ { 1 }\) The variance of \(\hat { \mathrm { p } } _ { 2 }\) is \(\frac { 7 p ( 1 - p ) } { 18 n }\)
3. State, giving a reason, which is the better estimator. The estimator \(\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }\) where \(a\) and \(b\) are positive integers.
4. Find the pair of values of \(a\) and \(b\) such that \(\hat { \mathrm { p } } _ { 3 }\) is a better unbiased estimator of \(p\) than both \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\)
   You must show all stages of your working.