Edexcel S2 (Statistics 2) 2004 January

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

The random variable \(R\) has the binomial distribution B(12, 0.35).

1. Find P(\(R \geq 4\)). [2]

The random variable \(S\) has the Poisson distribution with mean 2.71.

1. Find P(\(S \leq 1\)). [3]

The random variable \(T\) has the normal distribution N(25, \(5^2\)).

1. Find P(\(T \leq 18\)). [2]

The discrete random variable \(X\) is distributed 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. [1]
2. Give a reason to support your value. [1]
3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]

1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]

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,

1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]

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. [2]

Find the probability that in any randomly selected 10 minute interval

1. exactly 6 cars pass this point, [3]
2. at least 9 cars pass this point. [2]

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.

1. 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]

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\%. [5]
2. Write down the significance level of the above test. [1]

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.

1. 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 $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

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