AQA Further Paper 3 Statistics (Further Paper 3 Statistics) 2022 June

1 The random variable \(T\) follows a discrete uniform distribution and can take values \(1,2,3 , \ldots , 16\) Find the variance of \(T\) Circle your answer.
1.2518 .7521 .2521 .33

2 The random variable \(X\) has probability density function $$f ( x ) = \begin{cases} 1 & 0 < x \leq \frac { 1 } { 2 }
\frac { 3 } { 8 } x ^ { - 2 } & \frac { 1 } { 2 } < x \leq \frac { 3 } { 2 }
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X < 1 )\) Circle your answer.
[0pt] [1 mark]
\(\frac { 1 } { 8 }\)
\(\frac { 3 } { 8 }\)
\(\frac { 5 } { 8 }\)
\(\frac { 7 } { 8 }\)
\includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-03_2488_1718_219_153}

3 The random variable \(X\) has an exponential distribution with probability density function \(\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x }\) where \(x \geq 0\) 3

1. Show that the cumulative distribution function, for \(x \geq 0\), is given by \(\mathrm { F } ( x ) = 1 - \mathrm { e } ^ { - \lambda x }\)
   [0pt] [3 marks]
   3
2. Given that \(\lambda = 2\), find \(\mathrm { P } ( X > 1 )\), giving your answer to three decimal places.

4 Daisies and dandelions are the only flowers growing in a field. The number of daisies per square metre in the field has a mean of 16
The number of dandelions per square metre in the field has a mean of 10
The number of daisies per square metre and the number of dandelions per square metre are independent. 4

1. Using a Poisson model, find the probability that a randomly selected square metre from the field has a total of at least 30 flowers, giving your answer to three decimal places.
   4
2. A survey of the entire field is taken.
   The standard deviation of the total number of flowers per square metre is 10 State, with a reason, whether the model used in part (a) is valid.

5 The mass, \(X\), in grams of a particular type of apple is modelled using a normal distribution. A random sample of 12 apples is collected and the summarised results are $$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$ 5

1. A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample. Show that the confidence interval is \(( 81.7,91.3 )\) with values correct to three significant figures.
   5
2. Padraig claims that the population mean mass of the apples is 85 grams. He carries out a hypothesis test at the \(1 \%\) level of significance using the random sample of 12 apples. The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 85
   & \mathrm { H } _ { 1 } : \mu \neq 85 \end{aligned}$$ State, with a reason, whether the null hypothesis is accepted or rejected.
   5
3. Interpret, in context, the conclusion to the hypothesis test in part (b).

6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0
b & x = 1
c & x = 2
0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6

1. Deduce the values of \(a , b\) and \(c\)
   6
2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\)
   [0pt] [2 marks]

7

1. Test the scientist's claim, using the 10\% level of significance.
   7
2. For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]

8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0
\mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5
1 & x > 5 \end{cases}$$ 8

1. Show that \(k = \frac { 1 } { 5 } \ln 2\)
   [0pt] [2 marks]
   8
2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
   

   8	Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.

9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9

1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
   Determine whether Lianne will buy the battery.
   [0pt] [2 marks]
   9
2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged.
   \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges.
   \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
   \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}