Download g) Chapter 7 7.1 A confectionery company produces 100

g) Chapter 7
7.1
A confectionery company produces 100-gram ‘Real Chocolate’ bars that have a mean
chocolate content of 70 grams with a standard deviation of 0.8 gram. The variation in
chocolate content is normally distributed. What is the probability that a chocolate bar
chosen at random contains:
(a)
More than 71 grams?
(b)
More than 68 grams?
(c)
Less than 70 grams?
(d)
Between 69 and 72 grams?
7.2
The colour dye used in the manufacture of certain garments fades after a mean of 96
machine washes, with a standard deviation of 7 washes. If the number of washes before
fading follows a normal pattern, what proportion of the garments will first fade;
(a)
After 100 washes?
(b)
Before 90 washes?
(c)
Between 95 and 105 washes?
7.3
A bed linen company manufactures duvets that have a mean TOG rating (a measure of
thermal effectiveness) of 11.5 with a standard deviation of 0.18. If the TOG ratings are
normally distributed, what is the probability that a duvet selected at random has a TOG
rating:
(a)
Above 12?
(b)
Above 11.3?
(c)
Below 11.6?
(d)
Below 11.1?
(e)
Between 11.2 and 11.4?
(f)
Between 11.5 and 11.9?
7.4
The times taken by a traditional barber to cut hair are normally distributed with a mean of
15.6 minutes and a standard deviation of 3.4 minutes. What is the probability that he can
cut a customer’s hair in:
(a)
Less than 20 minutes?
(b)
Less than 10 minutes?
(c)
Between 8 and 14 minutes?
(d)
Between 17 and 22 minutes?
(e)
Between 12 and 18 minutes?
7.5
The amount of time that visitors to a web site browse the site is assumed to be normally
distributed with a mean of 8.75 minutes and a standard deviation of 3.1 minutes. What is
the probability that a randomly selected visitor browses the site for:
(a)
Less than 5 minutes?
(b)
More than 10 minutes?
(c)
Less than 15 minutes?
(d)
Between 3 and 7 minutes?
(e)
Between 10 and 14 minutes?
(f)
Between 8 and 12 minutes?
7.6
An automobile manufacturer produces a certain model of car. The fuel economy figures
of these cars are normally distributed with a mean mileage per gallon (mpg) of 36.8 and a
standard deviation of 1.3.
(a)
What is the probability that one of these cars will have an mpg of more than 37.5?
(b)
What is the probability that one of these cars will have an mpg of less than 35?
(c)
What is the probability that one of these cars will have an mpg of less than 40?
(d)
What is the probability that one of these cars will have an mpg of between 34 and
38 mpg?
(e)
What is the minimum mpg of the 15% most fuel efficient cars?
(f)
What is the maximum mpg of the 40% least fuel efficient cars?
7.7
A large company insists that all job applicants who are invited for interview take a
psychometric test. The results of these tests follow the normal distribution with a mean of
61 points and the standard deviation of 7.2 points.
(a)
What proportion of applications would be expected to score over 70 points?
(b)
What proportion of applications would be expected to score under 40 points
(c)
What proportion of applications would be expected to score between 50 and 65
points?
(d)
What score is exceeded by 20% of applicants?
(e)
What is the highest score achieved by the 5% of applicants who do least well in
the test?
7.8
The weight losses that members of a slimming club can expect to achieve follow the
normal distribution with a mean of 24.7 lb and a standard deviation of 5.2 lb.
(a)
What proportion of members can expect to lose more than 25 lb?
(b)
What proportion of members can expect to lose more than 20 lb?
(c)
What proportion of members can expect to lose less than 35 lb?
(d)
What proportion of members can expect to lose less than 15 lb?
(e)
What proportion of members can expect to lose between 12 lb and 30 lb?
(f)
What is the most weight that the least successful 40% of slimmers can expect to
lose?
(g)
What is the least weight that the most successful 30% of slimmers can expect to
lose?
7.9
A drinks vending machine dispenses a variety of hot and cold beverages. The volumes of
the drinks dispensed follow a normal pattern with a mean of 149.7 ml and a standard
deviation of 1.2 ml.
(a)
What proportion of drinks will be over 150 ml?
(b)
What proportion of drinks will be over 147.5 ml?
(c)
What proportion of drinks will be under 148 ml?
(d)
What proportion of drinks will be under 152 ml?
(e)
What proportion of drinks will be between 149 ml and 152.5 ml?
(f)
What is the most that can be expected in the smallest 5% of drinks?
(g)
What is the least that can be expected in the largest 25% of drinks?
7.10
The mean legal lifetime (the number of miles travelled before the tyre is worn down to
the legal limit) of car tyres of a certain brand is 23,450 miles. The standard deviation is
1,260 miles. If the lifetimes of the tyres are normally distributed, what is the probability
that a random sample of four tyres fitted to a vehicle will have a mean legal lifetime of:
(a)
More than 25,000 miles?
(b)
More than 22,000 miles?
(c)
Less than 24,000 miles?
(d)
Less than 23,000 miles?
(e)
Between 22,500 and 24,500 miles?
(f)
Between 23,400 and 24,200 miles?
7.11
As part of a charity event a radio DJ plans to broadcast a selection of 12 tracks chosen at
random from the very large stock of albums stored at the radio station. The lengths of the
tracks on these albums are normally distributed with a mean of 3 minutes 9 seconds and a
standard deviation of 43 seconds.
(a)
What is the probability that the mean track length of the selected sample is longer
than 3 minutes?
(b)
What is the probability that the mean track length of the sample of tracks selected
is shorter than 2 minutes 50 seconds?
(c)
What is the probability that the mean track length of the sample of tracks selected
is longer than 3 minutes 30 seconds?
(d)
What is the probability that the mean track length of the sample of tracks selected
is between 2 minutes 45 seconds and 3 minutes 15 seconds?
(e)
If the DJ has an hour for the show and 20 minutes are required for advertising
slots and messages, what is the probability that the show will run over time?
7.12
Multi-Purpose Vehicles (MPVs) produced by a manufacturer can accommodate seven
adults. The load capacity of the passenger compartment is 1240 lb. If the weights of
adults are distributed normally with a mean of 170 lb and a standard deviation of 18 lb,
what is the probability that the mean weight of a sample of seven adults will exceed the
maximum average weight that the load capacity permits?
7.13
The delays to scheduled airline departures at an international airport are known to follow
a skewed distribution with a mean of 11 minutes and a standard deviation of 7 minutes.
What is the probability that the mean delay of a random sample of 40 flights is:
(a)
More than 12 minutes?
(b)
More than 10 minutes?
(c)
Less than 8 minutes?
(d)
Less than 11 minutes?
(e)
Between 9 minutes and 10.5 minutes?
(f)
Between 11.5 and 12.5 minutes?
(g)
Between 9.5 and 14 minutes?
7.14
The amount of analgesic per pill in a proprietary brand of painkiller follows a normal
pattern with a mean of 7.5 mg. The pills are sold in bottles of 50. If the standard deviation
of the analgesic content of the 50 pills in one bottle is 0.4 mg, what is the probability that
the mean analgesic content of 50 pills is:
(a)
Above 7.35 mg?
(b)
Above 7.6 mg?
(c)
Below 7.55 mg?
(d)
Below 7.45 mg?
(e)
Between 7.4 and 7.65 mg?
7.15
The contents of pots of a certain brand of yoghurt are normally distributed with a mean of
101.4 grams. The yoghurts are sold in packs of four. The contents of each pot in one pack
of four were measured and the standard deviation was found to be 3.1 grams.
(a)
What mean amount of content will be exceeded by 10% of packs?
(b)
What mean amount of content will be exceeded by 5% of packs?
(c)
What mean amount of content will be exceeded by 1% of packs?
7.16
The times taken by buses to travel between two stops on a route follow a normal pattern
of variation with a mean of 13.9 minutes. A commuter makes this journey ten times each
week. One week she times the ten journeys and works out that the standard deviation of
them is 1.7 minutes. Treating the ten journeys in a week as a random sample:
(a)
What mean journey time will be exceeded one week in twenty?
(b)
What mean journey time will be exceeded one week in forty?
(c)
What mean journey time will not be exceeded 90% of the time?
7.17
Select the appropriate description for each term on the left hand side from the list on the
right-hand side.
(a)
the normal distribution
(i)
the z value for which P (Z > z) = 0.05
(b)
the Z distribution
(ii)
the estimated standard error
(c)
zα
(iii)
a distribution used for small samples
(d)
z0.05
(iv)
a symmetrical continuous distribution
(e)
a sampling distribution
(v)
the standard error
(f)
σ/√n
(vi)
the z value for which P (Z > z) = α
(g)
s/√n
(vii) the Standard Normal Distribution
(h)
a t distribution
(viii) shows how sample results vary