The proportion of teenagers in the UK population who are short sighted may be
taken to be 30%.
(a) Use the Binomial Distribution to calculate the probability that at least 3 out
of a random sample of 12 teenagers will be short sighted.
[3 pts]
(b) A family in which there are 3 teenagers is chosen at randomfrom all such
families in the UK. Give a reason why the use of a Binomial Distribution to
calculate the probability of at least 2 of the teenagers in the family being
short-sighted might not be appropriate.
(c) A random sample of 1000 teenagers is taken. Calculate the probability that
at least 340 will be short sighted.
[5 pts]
At each turn in a game, a player's score, denoted by the random variable x, is
found as follows.
The player throws a fair six-sided die once, AND if the number showing on the die
is 4,5 , or 6 then that number is the value of x.
If the number showing on the die is 1,2 , or 3 , then the player throws the die again
AND x is the number showing on the second throw.
(a) Show how the result P(x=6)=(1)/(4) is obtained.
(b) Show that the Var(x)=(113)/(48).
(c) The mean of a random sample of 200 values of x is denoted by ar{x} . Use the
Central Limit Theorem to estimate P(ar{x} >4).
A fruit grower uses a machine to sort apples into various grades. Grade C apples
have weights Uniformly Distributed in the interval 100 to 110 grams.
(a) Find the variance of the weight of a grade C apple.
(b) Ten randomly chosen grade C apples are packed in a bag.
Using the Central Limit Theorem, find an approximate value for the proba-
bility that the weight of the ten apples in the bag exceeds 1030 grams.
pts]
(c) The grower suspects that the machine is not working correctly and that the
mean weight, mu grams, of a grade C apple may be less than 105 grams.
Devise a test, at the 10% level of significance, based on the level of the apples
in five randomly chosen bags, each containing 10 apples, of the Null Hypothesis
mu =105, with the Alternative Hypothesis mu <105.
A study of a large sample of books by a particular author shows that the number of
words per sentence can be modeled by a Normal Distribution with mean 21.2 and
standard deviation7.3.
A researcher claims to have discovered a previously unknown book by this author.
The mean length of 90 sentences chosen at random in this book is found to be 19.4
words.
Assuming a population standard deviation of sentence lengths in this book is also
7.3 , test at the 5% level of significance whether the mean sentence length is the
same as the author's.
1. The proportion of teenagers in the UK population who are short sighted may be taken to be 30%.
(a) Use the Binomial Distribution to calculate the probability that at least 3 out of a random sample of 12 teenagers will be short sighted. [3 pts] (b) A family in which there are 3 teenagers is chosen at randomfrom all such families in the UK. Give a reason why the use of a Binomial Distribution to calculate the probability of at least 2 of the teenagers in the family being short-sighted might not be appropriate. [1 pt] (c) A random sample of 1000 teenagers is taken. Calculate the probability that at least 340 will be short sighted. [5 pts]
2. At each turn in a game, a player's score, denoted by the random variable X, is found as follows.
The player throws a fair six-sided die once, AND if the number showing on the die is 4, 5, or 6 then that number is the value of X.
If the number showing on the die is 1, 2, or 3, then the player throws the die again AND X is the number showing on the second throw.
x 123456 PX=
a) Show how the result P(X =6) = is obtained [1pt] (b) Show that the Var(X) = 113 [5 pts] (c) The mean of a random sample of 200 values of X is denoted by X. Use the Central Limit Theorem to estimate P(X > 4). [4 pts]
3. A fruit grower uses a machine to sort apples into various grades. Grade C apples have weights Uniformly Distributed in the interval 100 to 110 grams.
a) Find the variance of the weight of a grade C apple.
[3 pts]
(b) Ten randomly chosen grade C apples are packed in a bag
Using the Central Limit Theorem, find an approximate value for the proba- bility that the weight of the ten apples in the bag exceeds 1030 grams. 1] pts]
(c The grower suspects that the machine is not working correctly and that the mean weight, grams, of a grade C apple may be less than 105 grams.
Devise a test, at the 10% level of significance, based on the level of the apples in five randomly chosen bags, each containing 10 apples, of the Null Hypothesis = 105, with the Alternative Hypothesis < 105. [6 pts]
4. A study of a large sample of books by a particular author shows that the number of words per sentence can be modeled by a Normal Distribution with mean 21.2 and standard deviation7.3.
A researcher claims to have discovered a previously unknown book by this author. The mean length of 90 sentences chosen at random in this book is found to be 19.4 words.
Assuming a population standard deviation of sentence lengths in this book is also 7.3, test at the 5% level of significance whether the mean sentence length is the same as the author's. [5 pts]
