Download Statistical Distributions Review

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Statistical Distributions Review
1.
a)
b)
c)
Find P(Z≤1.5)
Find P(Z≥1.17)
Find P(−1.16≤Z≤1.32)
2.
In a country called Tallopia, the height of adults is normally distributed with a mean of 187.5
cm and a standard deviation of 9.5cm.
a)
What percentage of adults in Tallopia have a height greater than 197cm?
b)
A standard doorway in Tallopia is designed so that 99% of adults have a space of at
least 17 cm over their heads when going through a doorway. Find the height of a standard
doorway in Tallopia. Give your answer to the nearest cm.
3.
The heights of boys at a particular school follow a normal distribution with a standard
deviation of 5 cm. The probability of a boy being shorter than 153 cm is 0.705.
a)
Calculate the mean height of the boys.
b)
Find the probability of a boy being taller than 156 cm.
4.
A company manufactures television sets. They claim that the lifetime of a set is normally
distributed with a mean of 80 months and standard deviation of 8 months.
a)
What proportion of television sets break down in less than 72 months?
b)
(i) Calculate the proportion of sets which have a lifetime between 72 months and 90
months.
(ii) Illustrate this proportion by appropriate shading in a sketch of a normal
distribution curve.
c)
If a set breaks down in less than x months, the company replaces it free of charge.
They replace 4% of the sets. Find the value of x.
5.
The following table shows the probability distribution of a discrete random variable X.
x
0
2
5
9
P(X = x)
0.3
k
2k
0.1
a)
Find the value of k.
b)
Find E(X)
6.
Two boxes contain numbered cards as shown below.
Two cards are drawn at random, one from each box.
a)
Copy and complete the table below to show all nine equally likely outcomes.
3, 9
3, 10
3, 10
b)
Let S be the sum of the numbers on the two cards. Find the probability of each value
of S.
c)
Find the expected value of S.
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d)
Anna plays a game where she wins $50 if S is even and loses $30 if S is odd. Anna
plays the game 36 times. Find the amount she expects to have at the end of the 36 games.
7.
Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are labeled
1, 2, 3, 4. The score for each die is the number which lands face down.
a)
List the pairs of scores that give a sum of 6.
b)
The probability distribution for the sum of the scores on the two dice is shown below.
Find the value of p, of q, and of r.
c)
Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the
sum is 5 on three or more throw. Find the probability that Fred wins a prize.
8.
A factory makes switches. The probability that a switch is defective is 0.04. The factory tests
a random sample of 100 switches.
a)
Find the mean number of defective switches in the sample.
b)
Find the probability that there are exactly six defective switches in the sample.
c)
Find the probability that there is at least one defective switch in the sample.
9.
A company produces a large number of water containers. Each container has two parts, a
bottle and a cap. The bottles and caps are tested to check that they are not defective. A cap
has a probability of 0.012 of being defective. A random sample of 10 caps is selected for
inspection.
a)
Find the probability that exactly one cap in the sample will be defective.
b)
The sample of caps passes inspection if at most one cap is defective. Find the
probability that the sample passes inspection.
c)
(i) The heights of the bottles are normally distributed with a mean of 22 cm and a
standard deviation of 0.3 cm. Sketch a normal curve, shading the region representing where
the heights are less than 22.63 cm.
(ii) Find the probability that the height of a bottle is less than 22.63 cm.
d)
(i) A bottle is accepted if its height lies between 21.37 cm and 22.63 cm. Find the
probability that a bottle selected at random is accepted.
(ii) A sample of 10 bottles passes inspection if all of the bottles in the sample are
accepted. Find the probability that the sample passes inspection.
e)
The bottles and caps are manufactured separately. A sample of 10 bottles and a
sample of 10 caps are randomly selected for testing. Find the probability that both samples
pass inspection.