Edexcel S2 (Statistics 2)

1. Briefly explain what is meant by
   1. a statistical model,
      (2 marks)
   2. a sampling frame,
   3. a sampling unit.
   4. (a) Explain what is meant by the critical region of a statistical test.
   5. Under a hypothesis \(\mathrm { H } _ { 0 }\), an event \(A\) can happen with probability \(4 \cdot 2 \%\). The event \(A\) does then happen. State, with justification, whether \(\mathrm { H } _ { 0 }\) should be accepted or rejected at the \(5 \%\) significance level.

3

1. Briefly describe the main features of a binomial distribution. I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52 .
2. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\).
   (2 marks)
   After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\), find
3. the probability of getting no hearts,
4. the probability of getting 4 or more hearts.
5. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn.

4. A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:

1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution.
2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute.
3. Find the number of one-minute intervals, in the sample of 100 , in which more than 6 counts would be expected. \section*{STATISTICS 2 (A) TEST PAPER 10 Page 2}

1. A continuous random variable \(X\) has the cumulative distribution function

$$\begin{array} { l l } \mathrm { F } ( x ) = 0 & x < 2 ,
\mathrm {~F} ( x ) = k ( x - a ) ^ { 2 } & 2 \leq x \leq 6 ,
\mathrm {~F} ( x ) = 1 & x \geq 6 . \end{array}$$

1. Find the values of the constants \(a\) and \(k\).
2. Show that the median of \(X\) is \(2 ( 1 + \sqrt { 2 } )\).
3. Given that \(X > 4\), find the probability that \(X > 5\).

6. A small opinion poll shows that the Trendies have a \(10 \%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10 \%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.

1. At the \(5 \%\) significance level, test which is right, stating your null hypothesis carefully.
2. If it is indeed true that the Trendies are supported by \(55 \%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes.

7. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 6 x } { 175 } & 0 \leq x < 5
\mathrm { f } ( x ) = \frac { 6 x ( 10 - x ) } { 875 } & 5 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$

1. Verify that f is a probability density function.
2. Write down the probability that \(X < 1\).
3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains.
4. Find the probability that \(2 < X < 7\).