Đề thi giữa kỳ môn Xác suất thống kê| Trường Đại học Ngoại Thương

1 The distribution of the heights of some plants is normal and has a mean of
40 cm and a standard deviation of 2 cm. Find the probability that a randomly selected plant is (i) under 42 cm (ii) over 42 cm (iii) over 40 cm (iv) between 40 and 42 cm.
2 The distribution of the masses of some baby parrots is normal and has a
mean of 60 g and a standard deviation of 5 g. Find the probability that a randomly selected bird is (i) under 63 g (ii) over 63 g (iii) over 68 g between 63 and 68 g. (iv)
3 The distribution of the mass of sweets in a bag is normal and has a mean
of 100 g and a standard deviation 2 g. Find the probability that a randomly selected bag is (i) under 98 g (ii) over 98 g (iii) under 102 g (iv) between 98 and 102 g.
4 The distribution of the heights of 18-year-old girls may be modelled by the
normal distribution with mean 162.5 cm and standard deviation 6 cm. Find
the probability that the height of a randomly selected 18-year-old girl is (i) under 168.5 cm (ii) over 174.5 cm
(iii) between 168.5 and 174.5 cm.
5 A pet shop has a tank of goldfish for sale. All the fish in the tank were hatched
at the same time and their weights may be taken to be normally distributed
with mean 100 g and standard deviation 10 g. Melanie is buying a goldfish and
is invited to catch the one she wants in a small net. In fact the fish are much
too quick for her to be able to catch any particular fish, and the one which she
eventually nets is selected at random. Find the probability that its weight is (i) over 115 g (ii) under 105 g (iii) between 105 and 115 g.
6 When he makes instant coffee, Tony puts a spoonful of powder into a mug.
The weight of coffee in grams on the spoon may be modelled by the normal
distribution with mean 5 g and standard deviation 1 g. If he uses more than
6.5 g Julia complains that it is too strong and if he uses less than 4 g she tells
him it is too weak. Find the probability that he makes the coffee (i) too strong (ii) too weak (iii) all right.
7 A biologist finds a nesting colony of a previously unknown sea bird on a remote
island. She is able to take measurements on 100 of the eggs before replacing
them in their nests. She records their weights, w g, in this frequency table. Weight, w 25 Frequency 2 13 35 33 17 0
(i) Find the mean and standard deviation of these data.
(ii) Assuming the weights of the eggs for this type of bird are normally
distributed and that their mean and standard deviation are the same as
those of this sample, find how many eggs you would expect to be in each of these categories.
(iii) Do you think the assumption that the weights of the eggs are normally distributed is reasonable?
8 The length of life of a certain make of tyre is normally distributed about a
mean of 24 000 km with a standard deviation of 2500 km.
(i) What percentage of such tyres will need replacing before they have travelled 20 000 km?
(ii) As a result of improvements in manufacture, the length of life is still
normally distributed, but the proportion of tyres failing before 20 000 km is reduced to 1.5%.
(a) If the standard deviation has remained unchanged, calculate the new mean length of life.
(b) If, instead, the mean length of life has remained unchanged, calculate the new standard deviation.
9 A machine is set to produce nails of length 10 cm, with standard deviation
0.05 cm. The lengths of the nails are normally distributed.
(i) Find the percentage of nails produced between 9.95 cm and 10.08 cm in length.
The machine’s setting is moved by a careless apprentice with the consequence
that 16% of the nails are under 5.2 cm in length and 20% are over 5.3 cm.
(ii) Find the new mean and standard deviation.
10 The concentration by volume of methane at a point on the centre line of
a jet of natural gas mixing with air is distributed approximately normally
with mean 20% and standard deviation 7%. Find the probabilities that the concentration (i) exceeds 30% (ii) is between 5% and 15%.
(iii) In another similar jet, the mean concentration is 18% and the standard
deviation is 5%. Find the probability that in at least one of the jets the
concentration is between 5% and 15%.
11 In a particular experiment, the length of a metal bar is measured many times.
The measured values are distributed approximately normally with mean
1.340 m and standard deviation 0.021 m. Find the probabilities that any one measured value (i) exceeds 1.370 m
(ii) lies between 1.310 m and 1.370 m
(iii) lies between 1.330 m and 1.390 m.
(iv) Find the length l for which the probability that any one measured value is less than l is 0.1.
12 A factory produces a very large number of steel bars. The lengths of these
bars are normally distributed with 33% of them measuring 20.06 cm or more
and 12% of them measuring 20.02 cm or less.
Write down two simultaneous equations for the mean and standard deviation
of the distribution and solve to find values to 4 significant figures. Hence
estimate the proportion of steel bars which measure 20.03 cm or more.
The bars are acceptable if they measure between 20.02 cm and 20.08 cm.
What percentage are rejected as being outside the acceptable range?
13 The diameters D of screws made in a factory are normally distributed with
mean 1 mm. Given that 10% of the screws have diameters greater than
1.04 mm, find the standard deviation correct to 3 significant figures, and hence
show that about 2.7% of the screws have diameters greater than 1.06 mm.
Find, correct to 2 significant figures,
(i) the number d for which 99% of the screws have diameters that exceed d mm
(ii) the number e for which 99% of the screws have diameters that do not
differ from the mean by more than e mm.
14 A machine produces crankshafts whose diameters are normally distributed
with mean 5 cm and standard deviation 0.03 cm. Find the percentage of
crankshafts it will produce whose diameters lie between 4.95 cm and 4.97 cm.
What is the probability that two successive crankshafts will both have a diameter in this interval?
Crankshafts with diameters outside the interval 5 ± 0.05 cm are rejected. If
the mean diameter of the machine’s production remains unchanged, to what
must the standard deviation be reduced if only 4% of the production is to be rejected?
15 In a reading test for eight-year-old children, it is found that a reading score X
is normally distributed with mean 5.0 and standard deviation 2.0.
(i) What proportion of children would you expect to score between 4.5 and 6.0?
(ii) There are about 700 000 eight-year-olds in the country. How many
would you expect to have a reading score of more than twice the mean?
(iii) Why might educationalists refer to the reading score X as a ‘score out of 10’?
The reading score is often reported, after scaling, as a value Y which is
normally distributed, with mean 100 and standard deviation 15. Values of Y
are usually given to the nearest integer.
(iv) Find the probability that a randomly chosen eight-year-old gets a score, after scaling, of 103.
(v) What range of Y scores would you expect to be attained by the best 20% of readers?
16 Extralite are testing a new long-life bulb. The lifetimes, in hours, are
After extensive tests, they find that 19% of bulbs have a lifetime exceeding
5000 hours, while 5% have a lifetime under 4000 hours.
(i) Illustrate this information on a sketch.
(iii) Find the probability that a bulb chosen at random has a lifetime between 4250 and 4750 hours.
(iv) Extralite wish to quote a lifetime which will be exceeded by 99% of bulbs.
What time, correct to the nearest 100 hours, should they quote?
A new school classroom has six light-fittings, each fitted with an Extralite long-life bulb.
(v) Find the probability that no more than one bulb needs to be replaced
within the first 4250 hours of use.
17 Tyre pressures on a certain type of car independently follow a normal
distribution with mean 1.9 bars and standard deviation 0.15 bars.
(i) Find the probability that all four tyres on a car of this type have pressures
between 1.82 bars and 1.92 bars.
(ii) Safety regulations state that the pressures must be between 1.9 − b bars
and 1.9 + b bars. It is known that 80% of tyres are within these safety
limits. Find the safety limits.
18 The lengths of fish of a certain type have a normal distribution with mean
38 cm. It is found that 5% of the fish are longer than 50 cm.
(i) Find the standard deviation.
(ii) When fish are chosen for sale, those shorter than 30 cm are rejected. Find
the proportion of fish rejected.
(iii) 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm.
19 (i) The random variable X is normally distributed. The mean is twice the deviation.
observations are taken from this distribution, how many would you
20 In a certain country the time taken for a common infection to clear up is
of these infections clear up in less than 7 days.
In another country the standard deviation of the time taken for the infection
to clear up is the same as in part (i) but the mean is 6.5 days. The time taken is normally distributed.
(ii) Find the probability that, in a randomly chosen case from this country,
the infection takes longer than 6.2 days to clear up.
Student ID with the k ending do exercise k or k-5.
1 In a game five dice are rolled together.
(i) What is the probability that (a) all five show 1 (b) exactly three show 1 (c) none of them shows 1?
(ii) What is the most likely number of times for 6 to show?
2 A certain type of sweet comes in eight colours: red, orange, yellow, green,
blue, purple, pink and brown and these normally occur in equal proportions.
Veronica’s mother gives each of her children 16 of the sweets. Veronica says
that the blue ones are much nicer than the rest and is very upset when she
receives less than her fair share of them.
(i) How many blue sweets did Veronica expect to get?
(ii) What was the probability that she would receive fewer blue ones than she expected?
(iii) What was the probability that she would receive more blue ones than she expected?
3 In a particular area 30% of men and 20% of women are overweight and there
are four men and three women working in an office there. Find the probability that there are (i) 0 (ii) 1 (iii) 2 overweight men; (iv) 0 (v) 1 (vi) 2 overweight women;
(vii) 2 overweight people in the office.
What assumption have you made in answering this question?
4 On her drive to work Stella has to go through four sets of traffic lights.
She estimates that for each set the probability of her finding them red is 23 and
green 13. (She ignores the possibility of them being amber.) Stella also estimates
that when a set of lights is red she is delayed by one minute. (i) Find the probability of (a) 0 (b) 1 (c) 2
(d) 3 sets of lights being against her.
(ii) Find the expected extra journey time due to waiting at lights.
5 Pepper moths are found in two varieties, light and dark. The proportion of
dark moths increases with certain types of atmospheric pollution. At the time
of the question 30% of the moths in a particular town are dark. A research
student sets a moth trap and catches nine moths, four light and five dark.
(i) What is the probability of that result for a sample of nine moths?
(ii) What is the expected number of dark moths in a sample of nine?
The next night the student’s trap catches ten pepper moths.
(iii) What is the probability that the number of dark moths in this sample is
the same as the expected number?