AQA Further Paper 3 Statistics (Further Paper 3 Statistics) 2021 June

1 The discrete uniform distribution \(X\) can take values \(1,2,3 , \ldots , 10\)
Find \(\mathrm { P } ( X \geq 7 )\) Circle your answer. \(0.3 \quad 0.4 \quad 0.6 \quad 0.7\)

2 The random variable \(X\) has variance \(\operatorname { Var } ( X )\) Which of the following expressions is equal to \(\operatorname { Var } ( a X + b )\), where \(a\) and \(b\) are non-zero constants? Circle your answer.
[0pt] [1 mark]
\(a \operatorname { Var } ( X )\)
\(a \operatorname { Var } ( X ) + b\)
\(a ^ { 2 } \operatorname { Var } ( X )\)
\(a ^ { 2 } \operatorname { Var } ( X ) + b\)

3 In a game, it is only possible to score 10, 20 or 30 points. The probability of scoring 20 points is twice the probability of scoring 30 points.
The probability of scoring 20 points is half the probability of scoring 10 points.
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1. Find the mean points scored when the game is played once, giving your answer to two decimal places.
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2. Mina plays the game.
   Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.

4 Oscar is studying the daily maximum temperature in \({ } ^ { \circ } \mathrm { C }\) in a village during the month of June. He constructs a \(95 \%\) confidence interval of width \(0.8 ^ { \circ } \mathrm { C }\) using a random sample of 150 days. He assumes that the daily maximum temperature has a normal distribution.
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1. Find the standard deviation of Oscar's sample, giving your answer to three significant figures.
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2. Oscar calculates the mean of his sample to be \(25.3 ^ { \circ } \mathrm { C }\)
   He claims that the population mean is \(26.0 ^ { \circ } \mathrm { C }\)
   Explain whether or not his confidence interval supports his claim.
   4
3. Explain how Oscar could reduce the width of his 95\% confidence interval.

5 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x \leq 1
\frac { 1 } { 10 } x - \frac { 1 } { 10 } & 1 < x \leq 6
\frac { 1 } { 90 } x ^ { 2 } + \frac { 1 } { 10 } & 6 < x \leq 9
1 & x > 9 \end{array} \right.$$ 5

1. Find the probability density function \(\mathrm { f } ( x )\)
   5
2. Show that \(\operatorname { Var } ( X ) = \frac { 6737 } { 1200 }\)
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6 Danai is investigating the number of speeding offences in different towns in a country. She carries out a hypothesis test to test for association between town and number of speeding offences per year. 6

1. State the hypotheses for this test. 6
2. The observed frequencies, \(O\), have been collected and the expected frequencies, \(E\), have been calculated in an \(n \times m\) contingency table, where \(n > 3\) and \(m > 3\) One of the values of \(E\) is less than 5 6
3. 1. Explain what steps Danai should take before calculating the test statistic.
      6
4. (ii) State an expression for the test statistic Danai should calculate.
   6
5. Danai correctly calculates the value of the test statistic to be 45.22 The number of degrees of freedom for the test is 25
   Determine the outcome of Danai's test, using the \(1 \%\) level of significance.

7 The random variable \(X\) has an exponential distribution with parameter \(\lambda\) 7

1. Prove that \(\mathrm { E } ( X ) = \frac { 1 } { \lambda }\)
   7
2. Prove that \(\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }\)

8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
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2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
   The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
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3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places.
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