OCR S2 (Statistics 2) 2014 June

1 The random variable \(F\) has the distribution \(B ( 50,0.7 )\). Use a suitable approximation to find \(\mathbf { P } \boldsymbol { ( } \mathbf { F > } \mathbf { 4 0 } \boldsymbol { ) }\). [5]

2 The events organiser of a school sends out invitations to \(\mathbf { 1 5 0 }\) people to attend its prize day. From past experience the organiser knows that the number of those who will come to the prize day can be modelled by the distribution \(\mathbf { B } ( \mathbf { 1 5 0 } , \mathbf { 0 . 9 8 } )\).
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1. Explain why this distribution cannot be well approximated by either a normal or a Poisson distribution. [3]
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2. By considering the number of those who do not attend, use a suitable approximation to find the probability that fewer than 146 people attend. [4]

3 The random variable \(G\) has the distribution \(\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)\). One hundred observations of \(G\) are taken. The results are summarised in the following table.

1. By considering \(\mathrm { P } ( G < 40.0 )\), write down an equation involving \(\mu\) and \(\sigma\). [2]
2. Find a second equation involving \(\mu\) and \(\sigma\). Hence calculate values for \(\mu\) and \(\sigma\). [4]
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3. Explain why your answers are only estimates. [1]

4 A zoologist investigates the number of snakes found in a given region of land. The zoologist intends to use a Poisson distribution to model the number of snakes.
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1. One condition for a Poisson distribution to be valid is that snakes must occur at constant average rate. State another condition needed for a Poisson distribution to be valid. [1] Assume now that the number of snakes found in 1 acre of a region can be modelled by the distribution Po(4).
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2. Find the probability that, in 1 acre of the region, at least 6 snakes are found. [2]
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3. Find the probability that, in 0.77 acres of the region, the number of snakes found is either 2 or 3. [4]

5 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$

1. Show that this is a valid probability density function. [4]
2. Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
3. Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
4. Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
5. A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]

6 In a city the proportion of inhabitants from ethnic group \(\mathbf { Z }\) is known to be \(\mathbf { 0 . 4 }\). A sample of \(\mathbf { 1 2 }\) employees of a large company in this city is obtained and it is found that 2 of them are from ethnic group \(Z\). A test is carried out, at the \(5 \%\) significance level, of whether the proportion of employees in this company from ethnic group \(Z\) is less than in the city as a whole.
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1. State an assumption that must be made about the sample for a significance test to be valid. [1]
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2. Describe briefly an appropriate way of obtaining the sample. [2]
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3. Carry out the test. [7]
4. A manager believes that the company discriminates against ethnic group \(Z\). Explain whether carrying out the test at the 10\% significance level would be more supportive or less supportive of the manager's belief. [2]

7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by
\(n = 50\),
\(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\),
\(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
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1. Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
2. Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
3. Explain whether the assumption that times are normally distributed is necessary in answering
   (a) part (i),
   [0pt] (b) part (ii). [2]

8 The random variable \(W\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 3.60\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda < 3.60\). The test is based on a single observation of \(W\). The critical region is \(W = 0\).
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1. Find the significance level of the test. [2]
2. It is known that, when \(\boldsymbol { \lambda } = \boldsymbol { \lambda } _ { \mathbf { 0 } }\), the probability that the test results in a Type II error is \(\mathbf { 0 . 8 }\). Find the value of \(\lambda _ { 0 }\). [4] \section*{END OF QUESTION PAPER}