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

1 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.
[0pt] [1 mark]
0.10 .2523
\(2 \quad \mathrm {~A} \chi ^ { 2 }\) 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. 681215

3 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 \(n = 8\)
3

1. Find \(\mathrm { E } ( X )\)
   3
2. \(\quad\) Find \(\operatorname { Var } ( X )\)
   3
3. Find \(\mathrm { P } ( X \geq 6 )\)
   3
4. The dice was rolled 800 times and the results below were obtained.

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

   State, with a reason, how you would refine the model for the random variable \(X\).
   [0pt] [2 marks]

4 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 \(\pounds 28500\) and the standard deviation is \(\pounds 5100\)
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.
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5 The discrete random variable \(X\) has the following probability distribution. 5

1. Find \(\mathrm { P } ( X \leq 6 )\) 5
2. Let \(Y = 3 X + 2\)
   Show that \(\operatorname { Var } ( Y ) = 32.49\)
   5
3. The continuous random variable \(T\) is independent of \(Y\). Given that \(\operatorname { Var } ( T ) = 5\), find \(\operatorname { Var } ( T + Y )\)

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 45 } \left( x ^ { 3 } - 10 x ^ { 2 } + 29 x - 20 \right) & 1 \leq x \leq 4
0 & \text { otherwise } \end{array} \right.$$ 6

1. Find \(\mathrm { P } ( X < 2 )\)
   6
2. Verify that the median of \(X\) is 2.3 , correct to two significant figures.
   6
3. Find the mean of \(X\).

7 A restaurant has asked Sylvia to conduct a \(\chi ^ { 2 }\) test for association between meal ordered and age of customer. 7

1. State the hypotheses that Sylvia should use for her test. 7
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.
   7
3. State in context the correct conclusion to Sylvia's test.

8 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.
8

1. Investigate Stuart's claim, using a suitable test at the \(5 \%\) level of significance.
   8
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.
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