Edexcel S2 (Statistics 2) 2004 January

1. A large dental practice wishes to investigate the level of satisfaction of its patients.
   1. Suggest a suitable sampling frame for the investigation.
   2. Identify the sampling units.
   3. State one advantage and one disadvantage of using a sample survey rather than a census.
   4. Suggest a problem that might arise with the sampling frame when selecting patients.
   5. The random variable \(R\) has the binomial distribution \(\mathrm { B } ( 12,0.35 )\).
   6. Find \(\mathrm { P } ( R \geq 4 )\).
   The random variable \(S\) has the Poisson distribution with mean 2.71.
2. Find \(\mathrm { P } ( S \leq 1 )\). The random variable \(T\) has the normal distribution \(\mathrm { N } \left( 25,5 ^ { 2 } \right)\).
3. Find \(\mathrm { P } ( T \leq 18 )\).

3. The discrete random variable \(X\) is distributed \(\mathrm { B } ( n , p )\).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution.
2. Give a reason to support your value.
3. Given that \(n = 200\) and \(p = 0.48\), find \(\mathrm { P } ( 90 \leq X < 105 )\).

4. (a) Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. A researcher has suggested that 1 in 150 people is likely to catch a particular virus.
Assuming that a person catching the virus is independent of any other person catching it,
(b) find the probability that in a random sample of 12 people, exactly 2 of them catch the virus.
(c) Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus.

5. Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. Find the probability that in any randomly selected 10 minute interval
2. exactly 6 cars pass this point,
3. at least 9 cars pass this point. After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly.
   (6)

6. From past records a manufacturer of ceramic plant pots knows that \(20 \%\) of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.

1. Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20 . The probability of rejection in either tail should be as close as possible to \(2.5 \%\).
2. Write down the significance level of the above test. A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.
3. Using a \(5 \%\) level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period.

7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 56 }\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Evaluate \(\mathrm { E } ( X )\).
4. Find the modal value of \(X\).
5. Verify that the median value of \(X\) lies between 2.3 and 2.5.
6. Comment on the skewness of \(X\). Justify your answer. \section*{END}