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Normal distribution – exercise
1. A psychology student is investigating how motivation influences performance. She
records the times that rats which have recently been fed take to complete a maze, for
which an item of food is given as a reward. The times follow a Normal Distribution with
mean 46s (seconds) and standard deviation 8s.
i.
Find the probability that a randomly selected rat completes the maze in less than
40s.
ii.
Out of a sample of 30 rats, estimate the number expected to complete the maze in
a time between 45s and 55s.
iii.
Calculate times within which you would expect 95% of the rats to complete the
maze.
A second group of rats, that have been denied food for four hours, complete the maze.
Their results are:
43, 21, 46, 58, 31, 17, 55, 29, 22, 19
iv.
Calculate the mean and standard deviation of these results.
v.
Comment on the claim that the increased motivation of this second group of rats
has improved performance in general but decreased the consistency.
2. The daily sales of a certain high street computer shop selling tablets follow a Normal
distribution with mean £5,500 and standard deviation £475.
i.
Find the probability that on a randomly chosen day the sales are between £5,000
and £6,000.
ii.
Estimate the number of days in a month when the sales exceed £6,000.
iii.
Give two seasonal factors which might change your answers to i) and ii).
iv.
The manager of the shop wants to know why on some days the sales are
particularly low. She decides to analyse the circumstances of the days when the
lowest 10% of sales have been recorded. Find the values she needs to use for this.
3. A sports studies student is investigating the reaction times of elite sprinters. He obtains
data relating to the delay between the firing of the starting gun to the time that an
athlete’s foot leaves the starting block. These show that the times are approximately
Normally distributed with a mean of 0.165s and a standard deviation of 0.018s.
i.
Write down the probability that a randomly chosen athlete’s reaction time is
greater than 0.165s.
ii.
Find the probability that a randomly chosen athlete’s reaction time is between
0.163s and 0.166s.
iii.
Calculate times within which approximately two thirds of athletes will react.
iv.
Carrie, another student who is also a competitive sprinter, points out that reaction
time less than 0.1s is regarded as humanly impossible and would incur a false
start. Calculate the percentage of such times according to the model above.
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4. An environmental pressure group is comparing the lengths of fish caught in two lakes.
In Lake A, the length may be modelled by a Normal distribution with mean 28.1cm and
standard deviation 6.5cm.
i.
Write down the probability that a fish from this lake would have a length of exactly
29.5cm.
ii.
Find the expected number of fish, in a random sample of 50, with lengths between
20cm and 30cm.
iii.
In Lake B, the following lengths are measured for a random sample of fish:
19.8, 18.0, 22.8, 23.4, 28.7, 21.9, 19.2, 18.1, 23.4, 18.8, 25.6, 27.9
Calculate the mean and standard deviation of this data and make two
comparisons between the lengths of fish in the two lakes.
5. A Core Maths student is investigating the times taken for the pulse rates of students to
return to their resting value after periods of vigorous exercise.
i.
Write down three features of a Normal distribution.
ii.
In the first group studied, there were a significant number of A Level Dance
students. Explain how the distribution of results may be affected by this.
In a completely random selection of students the following recovery times
(in seconds) were observed.
78, 95, 77, 91, 116, 92, 92, 87, 76, 72, 89, 78, 80, 83, 87, 79, 67, 104,
111, 90, 88, 113, 108, 71, 88, 80, 85, 99, 82, 80, 74, 102, 87, 78, 69
iii.
Illustrate this data using a stem and leaf diagram.
iv.
By considering the shape of the distribution, comment on whether a Normal
distribution seems suitable.
v.
Calculate the mean and standard deviation of the results.
vi.
Calculate the percentage of results falling within one standard deviation of the
mean and explain whether this supports your answer to part iv).
6. The circumferences of male infants’ heads are Normally distributed. At the age of 3
years and 6 months, the mean circumference is 49.5cm and the standard deviation is
1.5cm.
i.
Explain why you would expect approximately two thirds of the circumferences to
lie between 48cm and 51cm.
ii.
Calculate values within which roughly 95% of circumferences would lie.
iii.
Find the percentage of infants of this age whose head circumference is between
49cm and 55.0cm.
iv.
Infants with the smallest 10% of circumferences are routinely checked for cognitive
development. Calculate the value for this.
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7. In a factory a machine dispenses spring water into bottles. The nominal volume of
each bottle is 1 litre. The volume of spring water dispensed can be adjusted, and, from
previous experience, follows a Normal distribution.
i.
Explain why it would not be satisfactory to set the mean volume dispensed to
exactly one litre.
In fact, the machine is set to dispense a volume of 1015ml. From previous data, the
standard deviation is known to be 4.6ml.
ii.
Calculate the probability that the volume dispensed is more than 1020ml.
iii.
The company would like the probability that a bottle is underfilled to be less than
1 in 1,000. Is this achieved?
iv.
Find the volume exceeded by 1% of the bottles.
After routine maintenance, it is suspected that the number of bottles being
overfilled (which can lead to bursting) has increased. The following random
sample of values is obtained:
1031, 1028, 1024, 1025, 1018, 1029, 1028, 1027, 1021, 1007, 1031, 1019
1016, 1021, 1017, 1021, 1004, 1036, 1007, 1022, 1024, 1019, 1020, 1026
v.
Illustrate this data using a box and whisker plot.
vi.
Calculate the mean and standard deviation of the data.
vii.
How do both these values help explain the increased overfilling?
8. The delays to domestic flights at a medium-sized regional airport may be modelled by
a Normal distribution with mean 12 minutes and standard deviation 2.5 minutes.
i.
Calculate the probability that a randomly selected domestic flight is delayed by
more than a quarter of an hour.
ii.
Explain how you would expect the delays to long-haul flights at an international
airport to differ.
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