A-Level Audit: Statistics

Instructions:
Answer as many questions as possible.
The questions become progressively more difficult.
Where answers are not rational, round your
response to either 2 or 3 decimal places.
NOTE: Entering Fractions and Powers
In this test there are some questions where the answers contain fractions or
powers. These questions will have a special input box like the one below,
which will allow you to enter the answers.
 If you have not used the special input boxes before, you can find an explanation about how to use them here. You need to upgrade your Flash PlayerGo to http://www.adobe.com/go/getflashplayer.

1. Classify each of the following as either a discrete (D) or a continuous (C) variable. (Write D or C in the box)

(i)    The number of cars in a car park.
(ii)   The reaction time of a driver.
(iii)  The age, in years, of a school pupil.

2. The mean of a set of numbers is 14 and the standard deviation is 5.

(a)    A new set of numbers is created by adding 4 to each number in the original set.

(i)   What will be the mean of the new set of numbers?

(ii)  What will be the standard deviation of the new set of numbers?

(b)    A new set of numbers is created by multiplying each of the numbers in the original set by 3.

(i)   What will be the mean of this new set of numbers?

(ii)  What will be the standard deviation of this new set of numbers?

3. The number of telephone calls made per day by a teacher was recorded for 7 days.
The data collected is listed below.

5, 4, 6, 7, 10, 3, 7

(a)    What is the mode?

(b)    What is the median?

(c)    Calculate the range.

(d)    Calculate the inter-quartile range.

(e)    Calculate the mean.
4. The lengths of the telephone calls made from a phone during one month are summarised below.

 Call Length (t, mins) Frequency 5 11 13 3 9

Estimate the median length of the phone calls.    mins

5. Calculate the standard deviation of this set of numbers.
1, 3, 4, 8, 9
6. A histogram is to be drawn for the data on waiting times given in the table below.
Complete the blank spaces in the table.

 Call Length (t, mins) Frequency 6 9 4 2 1 FrequencyDensity

7. The diagram shows a box and whisker plot which has been drawn using a data sample.

Are these statements true or false? (Write T or F in each box.)

(i)    The median of the sample is 7.

(ii)   The inter-quartile range of the sample is 4.

(iii)  The mean of the sample can be found from the diagram.

(iv)  The mode cannot be found from the diagram.

8. The probability of obtaining a 2 when a dice is rolled is .

A 2 is obtained and the dice is rolled again.

 What is the probability of now obtaining a 2? You need to upgrade your Flash PlayerGo to http://www.adobe.com/go/getflashplayer.
9. You toss a coin 5 times.

What is the probability that you obtain

10. Hannah buys a box of 10 light bulbs in a discount store.
The probability that a bulb does not work is 0.1.

(a)    Calculate the probability that all the bulbs work.

(b)    Calculate the probability that exactly one of the bulbs does not work.

(c)    Calculate the probability that at least 8 of the bulbs work.

11. The people in a room are classified according to their age and gender.
The results are listed in the table below.

A person is selected at random from the room.

A denotes the event that the person chosen is 21 or older.

B denotes the event that the person chosen is male.

A' denotes the event not A, and B' denotes the event not B.

Calculate the following probabilities.

12. When Ben and Jake play table tennis, the probability that Jake wins is .
They play 8 games of table tennis. X  is the number of games that Jake wins.

(a)    What is the mean of  X ?

(b)    What is the variance of  X ?
13. The number of text messages arriving at a mobile phone each hour follows a Poisson distribution with mean 4.

Find the probability that, in one hour,

(a)    no text messages arrive

(b)    one text message arrives

(c)    two or more text messages arrive.
14. A random variable X  follows a Poisson distribution with mean 10.

What is the variance of  X?
15. Would you expect these variables to have a normal distribution? (Write Y or N in each box.)

A     The heights of 6-year-old children.

B     The score when a fair dice is rolled.

C     The scores of children in an IQ test.

D     The number of heads that you get when you toss a coin.
16. A random variable, Z, is normally distributed with mean 0 and standard deviation 1.

Calculate the probabilites that:

(a)    Z > 1

(b)    Z > 2

(c)    Z < 2
17. The ages of the members of a sports club are randomly distributed with mean 36 and standard deviation 6.

Calculate the probability that a member chosen at random is

(a)    over 39

(b)    under 45

(c)    aged between 28.8 and 44.4.
18. The birds in a flock have masses that are normally distributed with mean 300 grams and standard deviation 80 grams.

(a)    What mass will be exceeded by 10% of the birds?    grams

(b)    What mass will be exceeded by 20% of the birds?    grams

(c)    What mass will be exceeded by 90% of the birds?    grams

19. On a large shingle beach there are pebbles which have masses that are normally distributed with mean 70 and standard deviation 45.

Samples made up of 25 pebbles are to be taken from the beach.

(a)    Which of these statements is correct?

A    The sample means will have a Poisson distribution.
B    The sample means will have a Binomial distribution.
C    The sample means will have a Normal distribution.

(b)    What will be the mean of the distribution of the sample means?

(c)    What will be the standard deviation of the distribution of the sample means?
20. Consider the three scatter plots shown below.

(a)    Which graph shows a positive correlation?    Graph

(b)    Which graph shows a negative correlation?    Graph

(c)    Which graph shows no correlation?    Graph

21. Calculate the product moment correlation coefficient for the data in this table

 X 1 2 3 4 Y 2 5 2 8

r =

22. A discrete variable, X,  has the probability shown in the table below.

 x 0 1 2 p(X = x) 0.5 0.3 0.2

(a)    Calculate the mean of X.

(b)    Calculate the variance of X.

23. A continuous random variable, X, has probability density function

Find k
24. A continuous random variable, X, has probability density function

Find the mean of  X.
25. A sample of fish is taken from a lake. It is known that the
lengths of the fish in the lake are normally distributed
with standard deviation 8 cm.

The lengths of the fish in the sample are

35cm, 47cm, 50cm, 60cm

Calculate a 95% confidence interval for the mean length of the fish in the lake.    ( , )