WJEC Further Unit 2 (Further Unit 2) 2024 June

1. Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.
   1. (i) Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
      (ii) Justify any distribution you have used in answering (a)(i).
   2. On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
      1. find the expected time he catches his first fish,
      2. calculate the probability that he will not catch a fish by 3 pm .
   3. On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
      [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
   4. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
      
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2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, \(X\) minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function. $$F ( x ) = \begin{cases} 0 & x < - 2
\frac { x + 2 } { 5 } & - 2 \leqslant x < 1
\frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2
1 & x > 2 \end{cases}$$

1. Calculate the upper quartile for the 'allowed time'.
2. Find \(f ( x )\), the probability density function, for all values of \(x\).
3. 1. Calculate the mean 'allowed time'.
   2. Interpret your answer in context.

1. A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean \(2 \cdot 2\).

1. Carry out a chi-squared goodness of fit test, using a \(10 \%\) significance level.
2. A chi-squared goodness of fit test for the Poisson distribution with mean \(2 \cdot 5\) is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.

4. An author poses the following question: Does using cash for transactions affect people's financial behaviour?
She collects data on 'Cash transactions as a \% of all transactions' and 'Household debt as a \(\%\) of net disposable income' from a random sample of 25 countries. The table below shows the data she collected. There are missing values, \(p\) and \(q\), for Malta and Denmark respectively. The summary statistics and scatter diagram below are for the other 23 countries. \begin{figure}[h] \end{figure} $$\begin{gathered} \sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435
\sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043 \end{gathered}$$

1. Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
2. Calculate the equation of the least squares regression line of Household debt as a \% of net disposable income \(( y )\) on Cash transactions as a \% of all transactions ( \(x\) ). The regression line \(x\) on \(y\) is given below. $$x = - 0 \cdot 24 y + 91 \cdot 92$$
3. By selecting the appropriate regression line in each case, estimate the values of \(p\) and \(q\) in the table.
4. Comment on the reliability of your answers in part (c).
5. Interpret the negative value of \(y\) for Mexico.

5. Lily is interested in the relationship between the way in which students learned Welsh and their attitude towards the Welsh language. Students were categorised as having learned Welsh in one of three ways:

• from one Welsh-speaking parent/carer at home,
• from two Welsh-speaking parents/carers at home,
• at school only, for those with no Welsh-speaking parents/carers at home.

The students were asked to rate their attitude towards the Welsh language from 'Very negative' to 'Very positive'. The following data for a random sample of 253 students were collected as part of a project. Lily intends to carry out a chi-squared test for independence at the \(5 \%\) level. She produces the following tables which are incomplete.

1. Calculate the values of \(F , G\) and \(H\).
2. Carry out Lily's chi-squared test for independence at the \(5 \%\) level.
3. By referring to the figures in the tables on pages 16 and 17, give two comments on the relationship between the way students learned Welsh and their attitude towards the Welsh language.

6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
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