MAS1401 Practical Exercises 5. [Assessed!]
Hand in your solutions by taking them to the Mathematics and
Statistics General Office on the 5th floor of Merz Court by 4pm on
Thursday 10th May.
1. This question is about the Binomial Distribution. Six crates of oranges are
delivered to a retailer. The probability each crate passes a quality inspection is
0.9, independent of the other crates.
(a) Without using Minitab, calculate the probability that all six of the
crates pass the inspection, and write this in your solutions, together
with the calculation you used to obtain it. [4 Marks]
(b) Now use Minitab [Calc>Probability Distributions>Binomial…] to
obtain the probability that exactly 5 of the crates pass the inspection,
and write this in your solutions. [2 Marks]
(c) Without using Minitab, use your answers to parts (a) and (b) to
calculate the probability that 4 or less of the crates pass the inspection.
Write this in your solutions, together with the calculation you used to
obtain it. [4 Marks]
2. This question is also about the Binomial Distribution, and uses Data Set 4 in
your own personalized collection, which can be found, along with the names
index, at:
http://www.mas.ncl.ac.uk/~ndw/mas1401.htm
A new method of preserving onions is supposed to be 80% successful. Your
Data Set 4 tells you the result when 500 onions have been preserved using the
method, and each one is successful independently of the others.
(a) Use Minitab [Stat>Basic Statistics>1 Proportion…] to obtain a 95%
confidence interval for the actual success rate. Write this in your
solutions. [4 Marks]
(b) Test the null-hypothesis that the success rate really is 80%. Write the
result of your test, with your interpretation, in your solutions.
[4 Marks]
3. At a particular location, the number of Sandworms per m2 of ground is
thought to follow a Poisson distribution with parameter λ=10.
(a) In your solutions, write down the mean and the standard deviation of
the number of sandworms you could expect to find in one square-metre
of ground at this location. [6 Marks]
(b) Use Minitab [Calc>Probability Distributions>Poisson…] to obtain
the probabilities of finding exactly: (i) 5 worms; (ii) 10 worms; (iii) 15
worms in one square-metre. Write these in your solutions. [6 Marks]
(c) Use Minitab to obtain the probabilities of finding: (i) less than 5
worms; (ii) more than 20 worms in one square metre. Write these in
your solutions.
[4 Marks]
(d) A fieldworker analyses one square metre of ground and finds 21
worms. Without using Minitab, but using information already
obtained in this question, test the null hypothesis that the Poisson
parameter really is equal to 10. State the result of the test, with your
interpretation, in your solutions.
[6 Marks]
P.T.O.
4. This is a more substantial question, and takes you through the procedure
involved in carrying out a goodness of fit test. In a famous experiment on
gene-expression by the scientist Mendel, the probability that the offspring of a
particular cross-breed of sweet-pea is red is hypothesized to be 0.25.
(a) If we take a sample of four offspring, and count the number which are
red, which type of probability distribution is an appropriate model for
the outcome r? Write the answer in your solutions. [2 Marks]
(b) Using Minitab, obtain the probabilities for each of the possible
outcomes r=0, r=1, …, r=4.
(NB: you should first list the possible outcomes in column C1 in your
Minitab worksheet. You then need to choose the appropriate
probability distribution from the Calc>Probability Distributions>
menu and use the “Input Column:” and “Optional Storage:” options
to store the probabilities in column C2).
Write these probabilities (to three decimal places) in your solutions.
[6 Marks]
(c) Suppose the experiment, in which the number of red offspring out of 4
is recorded, is carried out for 100 different plants. In column C3,
calculate the expected frequencies for each of the possible outcomes
r=0, r=1, …, r=4. Write these (to one decimal place) in your solutions.
[4 Marks]
(d) The observed frequencies for the no. of red offspring out of 4 are:
0: 24
1: 41
2: 23
3: 12
4: 0
Enter these (the frequencies only) into column C4, making sure they
match with your list of possible outcomes in C1.
Copy columns C3 and C4 into columns C6 and C7. (You can do this
using Data>Copy>Columns to Columns…, or directly from the
session window by typing “Copy C3 C4 C6 C7”). In columns C6 and
C7, carry out any pooling of categories necessary to make the
expected frequencies all be >5. In column C5, write down the scores
out of 4 which correspond to the new categories (NB. You will
probably want the last category to be a text entry rather than a
numerical entry. It’s a good idea to enter this first, or Minitab gets
confused!) Do not enter anything in your solutions. [0 Marks]
(e) In Column C8, obtain the quantities (O-E) 2/E which we will use
shortly. Write these quantities down in your solutions, indicating to
which category each one corresponds. [6 Marks]
(f) Obtain the test statistic X2, which is just the sum of all the terms (O-E)
2/E. (You can do this either by just typing the command “Sum C8”, or
by using Calc>Column Statistics… from the menu). Write this
answer in your solutions. [2 Marks]
(g) What is the appropriate value for the degrees of freedom, ν , for this
test? Answer in your solutions. [2 Marks]
(h) What is the appropriate probability distribution for carrying out the
goodness of fit test? Answer in your solutions. [2 Marks]
(i) Using the values of X2 and ν you obtained, and the appropriate
distribution from part (h), use Minitab to obtain the p-value for the test,
and write this in your solutions. Note that the p-value is obtained as
1 – (the cumulative probability). [6 Marks]
(j) Use the p-value to state your conclusions about the null hypothesis
H0: Pr (red) = 0.25 in your solutions. [4 Marks]
(k) In your solutions, identify the main difference between the outcome
from the experiment, and what would be expected under the null
hypothesis. [6 Marks]
[Total: 80 Marks]
Hand in your solutions by taking them to the Mathematics and
Statistics General Office on the 5th floor of Merz Court by 4pm on
Thursday 10th May.