AQA Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) 2020 June

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

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. Find a possible value for the degrees of freedom for the test. Circle your answer. [1 mark] 6 \quad 8 \quad 12 \quad 15

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.

   \(x\)	1	2	3	4	5	6	7	8
   Frequency	103	63	84	110	74	41	85	240

   State, with a reason, how you would refine the model for the random variable \(X\). [2 marks]

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]

The discrete random variable \(X\) has the following probability distribution.

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]

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]

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]

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]