WJEC Further Unit 5 (Further Unit 5) 2019 June

A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below. 32.1 \quad 29.9 \quad 31.0 \quad 31.1 \quad 32.5 \quad 30.8 \quad 30.7 \quad 31.5

1. Calculate a 95\% confidence interval for \(\mu\) based on this sample. [7]
2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations. [1]

The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.

1. Find the mean and the variance of \(X\). [2]
2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]

The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.

1. What is the largest acceptable standard deviation? [3]

The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.

1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [6]

Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder \(A\) was used by 12 players, randomly selected, and their mean weight gain was 900 g. Protein powder \(B\) was used by the other 10 players and their mean weight gain was 870 g. Let \(\mu_A\) and \(\mu_B\) be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder \(A\) and protein powder \(B\) respectively.

1. Calculate a 98\% confidence interval for \(\mu_A - \mu_B\). [4]
2. In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer. [2]
3. Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder \(A\) over protein powder \(B\). [4]
4. State one non-statistical assumption you have made in order to reach these conclusions. [1]

To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee must both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.

1. Explain why the Wilcoxon signed rank test is appropriate. [2]

The scores awarded are shown below.

1. 1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a 5\% significance level.
   2. What conclusion should be reached about the suitability of the trainee to qualify? [9]

A manufacturer of batteries for electric cars claims that an hour of charge can power a certain model of car to travel for an average of 123 miles. An electric car company and a consumer, Hopcyn, both wish to test the validity of the manufacturer's claim.

1. Explain why Hopcyn may want to use a one-sided test and why the car company may want to use a two-sided test. [2]

To test the validity of this claim, Hopcyn collects data from a random sample of 90 drivers of this model of car to see how far they travelled, \(X\) miles, on an hour of charge. He produced the following summary statistics. $$\sum x = 11007 \quad \sum x^2 = 1361913$$

1. 1. Assuming Hopcyn uses a one-sided test, state the hypotheses.
   2. Test at the 5\% significance level whether the manufacturer's claim is correct. [8]

Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers. Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28 New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15 Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]

The random variable \(X\) has probability density function $$f(x) = 1 + \frac{3\lambda x}{2} \quad \text{for } -\frac{1}{2} \leqslant x \leqslant \frac{1}{2},$$ $$f(x) = 0 \quad \text{otherwise,}$$ where \(\lambda\) is an unknown parameter such that \(-1 \leqslant \lambda \leqslant 1\).

1. 1. Find E\((X)\) in terms of \(\lambda\).
   2. Show that \(\text{Var}(X) = \frac{16 - 3\lambda^2}{192}\). [6]
2. Show that P\((X > 0) = \frac{8 + 3\lambda}{16}\). [2]

In order to estimate \(\lambda\), \(n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\overline{X}\).

1. 1. Identify the distribution of \(Y\).
   2. Show that \(T_1\) is an unbiased estimator for \(\lambda\), where $$T_1 = \frac{16Y}{3n} - \frac{8}{3}.$$ [4]
2. 1. Show that \(\text{Var}(T_1) = \frac{64 - 9\lambda^2}{9n}\).
   2. Given that \(T_2\) is also an unbiased estimator for \(\lambda\), where $$T_2 = 8\overline{X},$$ find an expression for Var\((T_2)\) in terms of \(\lambda\) and \(n\).
   3. Hence, giving a reason, determine which is the better estimator, \(T_1\) or \(T_2\). [6]