Download Paper 2 ST

ST BENEDICT’S COLLEGE
SUBJECT
AP Mathematics Paper
2
12 JULY 2016
STATISTICS
GRADE
EXAMINER
NAME
TEACHER
12
Mrs MH Povall
QUESTION NO
DESCRIPTION
MARKS
100
MODERATORS Mr N Benecke
DURATION
1 Hour
MAXIMUM
MARK
1
Probability
16
2
Probability
9
3
Combinations
6
4
Normal Distribution
17
Confidence Intervals
12
Probability Functions
14
Hypothesis Testing
14
Line of Regression
12
5
6
7
8
TOTAL
ACTUAL MARK
100
READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
1.
2.
3.
4.
5.
6.
This question paper consists of 7 pages and a formula sheet is supplied.
Read the questions carefully.
All the necessary working details must be clearly shown.
Approved non-programmable calculators may be used except where otherwise stated.
It is in your own interest to write legibly and to present your work neatly.
Where necessary, round off to four decimal places.
QUESTION 1
16 MARKS
Daniel and Tshepo are old boys of St Benedict’s and they sometimes attend the Saturday home sports
Fixtures.
The probability P(D), that Daniel attends a fixture is 0.70.
The probability P(T), that Tshepo attends a fixture is 0.55.
The probability that they both attend a fixture is 0.45.
a) Find the probability that either Daniel or Tshepo or both attend a fixture.
(3)
b) Find the probability that either Daniel or Tshepo but not both attend a fixture.
(2)
c) What is the probability that Daniel attends a fixture given that Tshepo is definitely going to
attend?
(3)
d) Give a numerical justification for the following statement:
“ Events D and T are not independent”
(4)
Jaryd and Craig are also old boys and they also sometimes attend the home fixtures.
The probability P(J), that Jaryd attends a fixture is 0.65.
The probability P(C), that Craig attends a fixture is 0.45.
e) If the attendance of all four of these old boys at a fixture is independent of one another, what is
the probability that all four attend a particular fixture?
(2)
f) What is the probability that none of them attend a particular fixture?
Grade 12
2 of 7
(2)
AP Mathematics Paper 2
QUESTION 2
9 MARKS
The probability that an online order from a supermarket chain has at least one item missing is 0.06.
Online orders are “incomplete” if they contain substitute items or have at least one item missing when
delivered. The probability that an order is incomplete is 0.15
a) Calculate the probability that exactly 2 of a random sample of 30 online orders have at least one item
missing when delivered.
(4)
b) Determine the probability that the number of incomplete orders in a sample of 50 online orders will
be more that 6 but fewer than 10.
(5)
QUESTION 3
6 MARKS
A mother goes shopping with her three children and buys 14 items. On her way home the shopping bag
breaks. She decides to carry five items herself, gives two to her youngest child, three to the middle child
and four to her eldest child. In how many ways can the 14 items be distributed in this manner?
Grade 12
3 of 7
AP Mathematics Paper 2
QUESTION 4
17 MARKS
a) The length of aluminium baking foil on a roll may be modelled by a normal distribution with a mean
of 91 metres and a standard deviation of 0.8 metres.
Determine the probability that the length of foil on a particular roll is:
i) less than 90 metres.
(6)
ii) not exactly 90 metres
(2)
iii) between 91 metres and 92.5 metres.
(4)
b) The length of cling wrap on a roll may also be modelled by a normal distribution but with a mean of
153 metres and a standard deviation of 𝜎 metres.
It is required that 1% of the rolls of cling wrap should have a length less that 150 meters.
Find the value of 𝜎 that is needed to satisfy this requirement.
Grade 12
4 of 7
(5)
AP Mathematics Paper 2
QUESTION 5
12 MARKS
In a sample of 400 shops it was discovered that 136 of them sold television sets below the list
price that has been recommended by the manufactures.
a) Calculate the 95% confidence interval for this estimate.
(7)
b) What size of sample would need to be taken in order to estimate the percentage within ±2%,
with the same confidence.
(5)
QUESTION 6
14 MARKS
A probability density function is defined as follows:
𝑞(4 − 𝑥); 0 ≤ 𝑥 ≤ 4
𝑓(𝑥) = {
for some constant q
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
a) Determine the value of q.
(5)
b) Calculate p(1 ≤ 𝑥 ≤ 2)
(4)
c) Calculate the median of this distribution.
(5)
Grade 12
5 of 7
AP Mathematics Paper 2
QUESTION 7
14 MARKS
A company produces low-energy light bulbs. The bulbs are described as using 9 watts of power,
with a standard deviation of 1.3 watts.
The production manager is concerned that the new machine being used on the production line has
changed this and he asks you to conduct a test at a 5% significance level to check whether his
concern is valid. You use a random sample of 120 bulbs of the latest batch produced and find
that the mean of the sample is 9.2 watts.
a) Conduct the hypothesis test and provide the conclusion you would give to the production
manager.
(8)
b) When you provide your conclusion, the production manager says that his intention was actually
to ask whether the bulbs in the batch use more than 9 watts since the new machine. Conduct
the hypothesis test in this case and provide the new conclusion you would give him.
Grade 12
6 of 7
(6)
AP Mathematics Paper 2
QUESTION 8
12 MARKS
The following data was collected during an experiment to determine the effect of temperature,
(𝑥° C) on the pH (y) of soya milk.
a) By observing the above data, explain what is revealed about the relationship between 𝑥 and 𝑦.
(1)
b) Use your calculator to evaluate the correlation coefficient r for this data and make a conclusion
about the strength of the straight line relationship.
(3)
c) Your calculator is faulty and won’t allow you to find the line of regression directly (ie it won’t let
you calculate a and b.) However, it will allow you to use the sum function. Use this information
to calculate the value of b for this data and hence calculate the equation of the line of best fit
showing all of your working out.
(5)
d) i) Determine the pH of soya milk at 95°C.
(1)
ii) Indicate, with a reason, how reliable this estimate might be.
Grade 12
7 of 7
(2)
AP Mathematics Paper 2