Questions — AQA Further AS Paper 2 Statistics (60 questions)

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AQA Further AS Paper 2 Statistics 2024 June Q6
11 marks Challenging +1.2
6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5 \\ 0 & \text { otherwise } \end{cases}$$ 6
  1. Find \(\mathrm { P } ( X > 2 )\) [0pt] [2 marks]
    6
  2. Find the upper quartile of \(X\) Give your answer to two decimal places.
    6
  3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.
AQA Further AS Paper 2 Statistics 2024 June Q7
11 marks Standard +0.3
7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7
  1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
    7
  2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
    Fully justify your answer.
    7
  3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
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AQA Further AS Paper 2 Statistics 2020 June Q1
1 marks Easy -1.8
The discrete random variable \(X\) has the following probability distribution function. $$\mathrm{P}(X = x) = \begin{cases} 0.2 & x = 1 \\ 0.3 & x = 2 \\ 0.1 & x = 3, 4 \\ 0.25 & x = 5 \\ 0.05 & x = 6 \\ 0 & \text{otherwise} \end{cases}$$ Find the mode of \(X\). Circle your answer. [1 mark] 0.1 \quad 0.25 \quad 2 \quad 3
AQA Further AS Paper 2 Statistics 2020 June Q2
1 marks Moderate -0.8
A \(\chi^2\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test.
Number of times late
01234
A8.121415.12144.76
Class B8.9915.516.7415.55.27
C11.8920.522.1420.56.97
Find a possible value for the degrees of freedom for the test. Circle your answer. [1 mark] 6 \quad 8 \quad 12 \quad 15
AQA Further AS Paper 2 Statistics 2020 June Q3
5 marks Moderate -0.8
The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\)
  1. Find E\((X)\) [1 mark]
  2. Find Var\((X)\) [1 mark]
  3. Find P\((X \geq 6)\) [1 mark]
  4. The dice was rolled 800 times and the results below were obtained.
    \(x\)12345678
    Frequency1036384110744185240
    State, with a reason, how you would refine the model for the random variable \(X\). [2 marks]
AQA Further AS Paper 2 Statistics 2020 June Q4
3 marks Moderate -0.8
Murni is investigating the annual salary of people from a particular town. She takes a random sample of 200 people from the town and records their annual salary. The mean annual salary is £28 500 and the standard deviation is £5100 Calculate a 97% confidence interval for the population mean of annual salaries for the people who live in the town, giving your values to the nearest pound. [3 marks]
AQA Further AS Paper 2 Statistics 2020 June Q5
7 marks Moderate -0.3
The discrete random variable \(X\) has the following probability distribution.
\(x\)2469
P\((X = x)\)0.20.60.10.1
  1. Find P\((X \leq 6)\) [1 mark]
  2. Let \(Y = 3X + 2\) Show that Var\((Y) = 32.49\) [5 marks]
  3. The continuous random variable \(T\) is independent of \(Y\). Given that Var\((T) = 5\), find Var\((T + Y)\) [1 mark]
AQA Further AS Paper 2 Statistics 2020 June Q6
8 marks Standard +0.3
The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{4}{45}(x^3 - 10x^2 + 29x - 20) & 1 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. Find P\((X < 2)\) [2 marks]
  2. Verify that the median of \(X\) is 2.3, correct to two significant figures. [4 marks]
  3. Find the mean of \(X\). [2 marks]
AQA Further AS Paper 2 Statistics 2020 June Q7
6 marks Moderate -0.8
A restaurant has asked Sylvia to conduct a \(\chi^2\) test for association between meal ordered and age of customer.
  1. State the hypotheses that Sylvia should use for her test. [1 mark]
  2. Sylvia correctly calculates her value of the test statistic to be 44.1 She uses a 5% level of significance and the degrees of freedom for the test is 30 Sylvia accepts the null hypothesis. Explain whether or not Sylvia was correct to accept the null hypothesis. [4 marks]
  3. State in context the correct conclusion to Sylvia's test. [1 mark]
AQA Further AS Paper 2 Statistics 2020 June Q8
9 marks Challenging +1.2
There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day. Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time. On a randomly chosen day, he records a total of 16 births from the two hospitals.
  1. Investigate Stuart's claim, using a suitable test at the 5% level of significance. [6 marks]
  2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. [3 marks]