Question 1
In
the following table, X represents the number of calls for NRMAâs road
assistance on a given day in a small university town.
X
5
6
7
8
9 10
P(X)
0.15 0.18
0.34 0.16 0.12
0.05
(a)
Express the probability distribution in a graphical form.
(b)
What is the expected number of calls per day?
(c)
What is the standard deviation of the number of calls per day?
(d)
What is the probability that more than eight calls will be made in a given day?
(e)
What is the probability that at least seven calls will be made in a given day?
Question 2
A
new cafe succeeds 70% of the time. Suppose that 3 such businesses open in
Armidale. Assume that their relative successes would be independent.
(a)
Show that the probability that all 3 businesses succeed is 0.343. You may use
either a probability tree or the binomial formula to answer this question.
(b)
For questions (i) to (iii), use the binomial table below:
(i) What is the
probability that all 3 businesses fail?
(ii) What is the
probability that at least 1 business succeeds?
(iii) What is the probability
that exactly 1 business succeeds?
(c)
Calculate the mean and standard deviation of the number of businesses that will
succeed.
Question 3
Let
X be a normal random variable with a mean of 70 and a standard deviation of
3. Calculate the following values. Show all steps in your answers.
(a)
P(X ? 67)
(b)
P(X ? 72)
(c)
P(X ? 75)
(d)
P(67 < X? 75)
(e)
If P(X ? a) = 0.95, find the value of a.
(f)
70% of the values are above which X value?
(g)
10% of the values are less than which X value?
Question 4
Suppose
that the actual amount of instant coffee that a filling machine puts into 150gm
jars varies from jar to jar. The actual fill may be considered to be a random
variable having a normal distribution with a mean of 150gms and a standard
deviation of 1gm.
(a)
What is the probability that a randomly selected coffee jar contains less
than 148gms of coffee?
(b)
What is the probability that a randomly selected coffee jar contains more
than 151.5gms of coffee?
(c)
What is the probability that a randomly selected coffee jar contains between
148gms and 150.5gms of coffee?
(d)
The heaviest 5% of jars are determined to be over-weight. At what weight is a
jar deemed to be overweight?
Question 5
An
orange juice producer buys all his oranges from a large orange orchard. The
amount of juice squeezed from each orange is normally distributed with a mean
of 110 grams and a standard deviation of 18 grams. Suppose a sample of 36
oranges is selected.
(a)
What distribution does the sample mean of orange juice have?
Explain.
(b)
What is the probability that the sample mean will be at least 107 grams?
(c)
What is the probability that the sample mean will be less than 115
grams?
(d)
What is the probability that the sample mean will be between 106 and 112
grams?
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