OCR MEI S3 (Statistics 3) 2008 June

1. Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by $$f(x) = k(20-x) \text{ for } 0 \leq x \leq 20, \text{ where } k \text{ is a constant.}$$
   1. Find \(k\). Sketch the graph of \(f(x)\) and use its shape to explain what can be deduced about how long Sarah has to wait. [5]
   2. Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus. [4]
   3. Find the median length of time that Sarah has to wait. [3]
2. 1. Define the term 'simple random sample'. [2]
   2. Explain briefly how to carry out cluster sampling. [3]
   3. A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. [2]

An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.

1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]

1. A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.

   Site	A	B	C	D	E	F	G	H
   Type I	225.2	268.9	303.6	244.1	230.6	202.7	242.1	247.5
   Type II	215.2	242.1	260.9	241.7	245.5	204.7	225.8	236.0

   1. The grower intends to perform a \(t\) test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption. [3]
   2. Carry out the test using a 5\% significance level. [7]
2. The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.

   Tea	Q	R	S	T	U	V	W	X	Y	Z
   Taster 1	69	79	85	63	81	65	85	86	89	77
   Taster 2	74	75	99	66	75	64	96	94	96	86

   Use a Wilcoxon test to examine, at the 5\% level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ. [8]

1. A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.

   Number of bees	0	1	2	3	4	5	6	7	\(\geq 8\)
   Number of intervals	6	16	19	18	17	14	6	4	0

   1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer. [3]
   2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data. [10]
2. The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a 95\% confidence interval for the overall mean time. [4]