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extension questions

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.

Find the probability that a randomly selected rat completes the maze in less than 40s.
Out of a sample of 30 rats, estimate the number expected to complete the maze in a time between 45s and 55s.
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

Calculate the mean and standard deviation of these results.
Comment on the claim that the increased motivation of this second group of rats has improved performance in general but decreased the consistency.

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The daily sales of a certain high street computer shop selling tablets follow a Normal distribution with mean £5,500 and standard deviation £475.

Find the probability that on a randomly chosen day the sales are between £5,000 and £6,000.
Estimate the number of days in a month when the sales exceed £6,000.
Give two seasonal factors which might change your answers to i) and ii).
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.

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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.

Write down the probability that a randomly chosen athlete’s reaction time is greater than 0.165s.
Find the probability that a randomly chosen athlete’s reaction time is between 0.163s and 0.166s.
Calculate times within which approximately two thirds of athletes will react.

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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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.

Write down the probability that a fish from this lake would have a length of exactly 29.5cm.
Find the expected number of fish, in a random sample of 50, with lengths between 20cm and 30cm.
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.

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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.

Write down three features of a Normal distribution.
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

Illustrate this data using a stem and leaf diagram.
By considering the shape of the distribution, comment on whether a Normal distribution seems suitable.
Calculate the mean and standard deviation of the results.
Calculate the percentage of results falling within one standard deviation of the mean and explain whether this supports your answer to part iv).

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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.

Explain why you would expect approximately two thirds of the circumferences to lie between 48cm and 51cm.
Calculate values within which roughly 95% of circumferences would lie.
Find the percentage of infants of this age whose head circumference is between 49cm and 55.0cm.
Infants with the smallest 10% of circumferences are routinely checked for cognitive development. Calculate the value for this.

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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.

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.

Calculate the probability that the volume dispensed is more than 1020ml.
The company would like the probability that a bottle is underfilled to be less than �1 in 1,000. Is this achieved?
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

Illustrate this data using a box and whisker plot.
Calculate the mean and standard deviation of the data.
How do both these values help explain the increased overfilling?

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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.

Calculate the probability that a randomly selected domestic flight is delayed by more than a quarter of an hour.
Explain how you would expect the delays to long-haul flights at an international airport to differ.

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