Edexcel FS2 (Further Statistics 2) 2020 June

1 Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\). Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a \(5 \%\) level of significance and state your hypotheses clearly.

2 Jemima makes jam to sell in a local shop. The jam is sold in jars and the weight of jam in a jar is normally distributed. Jemima takes a random sample of 8 of her jars of jam and weighs the contents of each jar, \(x\) grams. Her results are summarised as follows $$\sum x = 3552 \quad \sum x ^ { 2 } = 1577314$$

1. Calculate a 95\% confidence interval for the mean weight of jam in a jar. The labels on the jars state that the average contents weigh 440 grams.
2. State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam.

3 Below are 3 sketches from some students of the residuals from their linear regressions of \(y\) on \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_252_704_342_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_266_718_625_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_248_599_936_660} \section*{III} III For each sketch you should state, giving your reason,

1. whether or not the sketch is feasible
   and if it is feasible
2. whether or not the sketch suggests a linear or a non-linear relationship between \(y\) and \(x\).

4 A biased coin has a probability \(p\) of landing on heads, where \(0 < p < 1\) Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads. Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of \(p\).
Simon proposes the estimator \(S = \frac { X + Y } { m + n }\) and Taruni proposes \(T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]\)

1. Show that both \(S\) and \(T\) are unbiased estimators of \(p\).
2. Prove that, for all values of \(m\) and \(n , S\) is the better estimator.

5 \begin{figure}[h] \end{figure} The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi
0 & \text { otherwise } \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 2 \pi }\) The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\)
   The probability density function of \(Y\) is \(g ( y )\), where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\)
2. find the exact value of \(\sigma\)
3. Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)

6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a \(95 \%\) confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the \(t\) distribution to obtain a \(95 \%\) confidence interval of \(( 1.193,1.367 )\) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

1. Test, at the \(10 \%\) level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.

7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .

1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.

8 A circle, centre \(O\), has radius \(x \mathrm {~cm}\), where \(x\) is an observation from the random variable \(X\) which has a rectangular distribution on \([ 0 , \pi ]\)

1. Find the probability that the area of the circle is greater than \(10 \mathrm {~cm} ^ { 2 }\)
2. State, giving a reason, whether the median area of the circle is greater or less than \(10 \mathrm {~cm} ^ { 2 }\) The triangle \(O A B\) is drawn inside the circle with \(O A\) and \(O B\) as radii of length \(x \mathrm {~cm}\) and angle \(A O B x\) radians.
3. Use algebraic integration to find the expected value of the area of triangle \(O A B\). Give your answer as an exact value.