SPS SPS FM Statistics (SPS FM Statistics) 2024 September

1. The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{a65400d1-fadc-4bc7-ba4b-af2df57e390a-04_551_894_395_169} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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2.

1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls. Find the probability that the team contains no more than 3 girls.
2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).
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3. The random variable \(D\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\operatorname { Var } ( D ) = \frac { 40 } { 9 }\).
Determine

1. \(\operatorname { Var } ( 3 D + 5 )\),
2. \(\mathrm { E } ( 3 D + 5 )\),
3. \(\mathrm { P } ( D > \mathrm { E } ( D ) )\).
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4. The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
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5. At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
2. Find
   (a) \(\mathrm { P } ( X = 3 )\),
3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
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6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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7. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(2 a + 2 b = 1\).
2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).
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