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
$$\mathrm { f } ( x ) = k ( 20 - x ) \text { for } 0 \leqslant x \leqslant 20 \text {, where } k \text { is a constant. }$$
Find \(k\). Sketch the graph of \(\mathrm { f } ( x )\) and use its shape to explain what can be deduced about how long Sarah has to wait.
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.
Find the median length of time that Sarah has to wait.
Define the term 'simple random sample'.
Explain briefly how to carry out cluster sampling.
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.
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.
Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms.
Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms.
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.
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.
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.
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
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.
Carry out the test using a \(5 \%\) significance level.
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.
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
\(\geqslant 8\)
Number of intervals
6
16
19
18
17
14
6
4
0
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.
Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
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.