WJEC Further Unit 5 (Further Unit 5) 2024 June

1. During practice sessions, a basketball coach makes his players run several 'line drills'.
   1. He records the times taken, in seconds, by one of his players to run the first 'line drill' on a random sample of 8 practice sessions. The results are shown below.
      \(\begin{array} { l l l l l l l l } 29.4 & 31.1 & 28.9 & 30.0 & 29.9 & 30.4 & 29.7 & 30.2 \end{array}\)
      Assuming that these data come from a normal distribution with mean \(\mu\) and variance \(0 \cdot 6\), calculate a \(95 \%\) confidence interval for \(\mu\).
   2. State the two ways in which the method used to calculate the confidence interval in part (a) would change if the variance were unknown.
   3. During a practice session, a player recorded a mean time of 35.6 seconds for 'line drills'.
      1. Give a reason why this player may not be the same as the player in part (a).
      2. Give a reason why this player could be the same as the player in part (a).
   4. In country \(A\), the median daily caffeine intake per student who drinks coffee is 120 mg . A university professor who oversees a foreign exchange programme believes that students visiting from country B drink more coffee and therefore have a greater daily caffeine intake from coffee.
   On a randomly chosen day, the caffeine intake, in mg , from coffee consumption by each of 15 randomly selected students from country B is given below.

   136	149	202	0	110	0	100	180
   0	187	0	0	138	197	115

   The professor suspects that the students with zero caffeine intake do not drink coffee, and decides to ignore those students and instead focus on the coffee-drinking students.
2. Conduct an appropriate Wilcoxon test at a significance level as close to \(5 \%\) as possible. State your conclusion in context.
3. State one limitation of this investigation.

3. Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.

1. Calculate an approximate \(90 \%\) confidence interval for \(p\), the probability that a randomly selected customer buys a steak pie.
2. Suppose that Tony carries out 50 such surveys and calculates \(90 \%\) confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of \(p\).

4. The sports performance director at a university wishes to investigate whether there is a difference in the means of the specific gravities of blood of cyclists and runners. She models the distribution of specific gravity for cyclists as \(\mathrm { N } \left( \mu _ { X } , 8 ^ { 2 } \right)\) and for runners as \(\mathrm { N } \left( \mu _ { Y } , 10 ^ { 2 } \right)\).

1. State suitable hypotheses for this investigation.
   The mean specific gravity of blood of a random sample of 40 cyclists from the university was 1063. The mean specific gravity of blood of a random sample of 40 runners from the same university was 1060.
2. Calculate and interpret the \(p\)-value for the data.
3. Suppose now that both samples were of size \(n\), instead of 40. Find the least value of \(n\) that would ensure that an observed difference of 3 in the mean specific gravities would be significant at the \(1 \%\) level.

5. The probability density function of the continuous random variable \(X\) is given by $$\begin{array} { l l } f ( x ) = \frac { 3 x ^ { 2 } } { \alpha ^ { 3 } } & \text { for } 0 \leqslant x \leqslant \alpha
f ( x ) = 0 & \text { otherwise. } \end{array}$$ \(\bar { X }\) is the mean of a random sample of \(n\) observations of \(X\).

1. 1. Show that \(U = \frac { 4 \bar { X } } { 3 }\) is an unbiased estimator for \(\alpha\).
   2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\) ?
2. \(\quad \bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4 \bar { X } _ { 1 } - \frac { 8 } { 3 } \bar { X } _ { 2 }\) is also an unbiased estimator for \(\alpha\).
   1. Show that \(\frac { \operatorname { Var } ( U ) } { \operatorname { Var } ( V ) } = \frac { 1 } { 13 }\).
   2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator.

6. Alana is a PhD student researching language acquisition. She gives one group of randomly selected participants, Group A, 4 minutes to memorise 40 words that are similar in meaning. She gives a different, randomly selected group of participants, Group B, 4 minutes to memorise 40 words that are different in meaning. Alana believes that the students in Group B will do better than the students in Group A. The following results are the number of words recalled on testing the students from the two groups. Conduct a Mann-Whitney U test at a significance level as close as possible to \(5 \%\) to test Alana's belief.

7. A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.

1. The mass \(W\), in kg , of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation \(0 \cdot 6\). Given that, for containers of wheat, \(10 \%\) store less than 19 kg , find the value of \(\mu\).
   The mass \(X\), in kg , of corn stored in each individual container is normally distributed with mean \(20 \cdot 1\) and standard deviation \(1 \cdot 2\).
2. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg .
   The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean \(22 \cdot 2\) and standard deviation \(1 \cdot 5\). The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.
3. Calculate the probability that the farmer's wife will move
   1. the einkorn,
   2. the corn.
4. The mass \(E\), in kg , of emmer stored in each individual container is normally distributed with mean \(10 \cdot 5\) and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208 .
   1. Find the value of \(\sigma\) that the farmer's son used.
   2. Explain why the value of \(\sigma\) that he used is unreasonable.
      Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}