Question 1:
A soft drink manufacturer is using a "winning" cap as a sales promotion. The purchaser can get a prize if they buy a bottle that has a "Winner" symbol under the cap. Winning symbol caps are placed randomly on 5% of the bottles.
You buy a six-pack of the soft drink.
a) State the exact probability distribution that applies to this circumstance.
b) What is the probability that you win something? Give clear notation of the probability calculated.
c) What is the Poisson approximation to this distribution? Estimate the probability that you win something using this approximation.
d) Comment on the results of the two calculations.
Question 2:
The scores on an exam are normally distributed with µ = 30 and s = 7. (Show your working.)
a) What percent of the population have scores above 40?
b) What percent have scores between 30 and 40?
c) What score will place a student in the top 5% (to 1 decimal place)?
Question 3:
In a given large town the number of children, X, in a family chosen at random has the following distribution:
x P(X=x)
0 0.2
1 0.4
2 0.25
3 0.1
4 0.05
a) Find the mean and variance of X.
b) State the approximate distribution for the total number of children in 150 families.
On what theoretical basis can this approximation be made (state the reason)?
c) If a sample of 150 families is chosen at random what are the approximate probabilities there will be
i. more than 250 children present
ii. between 220 and 250 children present.
d) When catering for the children of these 150 families the organisers of a picnic for them wish to be 95% certain that they have provided enough food for the children.
How many children should they cater for?
Question 4:
A manufacturer of squash racquets has collected information on time in months to breakage for their racquets in normal play; that is, not due to mishandling. Data from past records on breakages suggest an average of 17.5 months and a standard deviation of 6.2 months.
Assume a normal distribution of time to breakage.
a) What proportion of racquets last at least two years?
b) The manufacturer offers a guarantee period in which racquets breaking in normal playing conditions are replaced. What guarantee period should be set if it is desired to limit the probability of replacing a racquet to 0.01?
c) Racquets can be bought in packs of two. Assuming that the second racquet is not used until the first is broken, what proportion of packs last at least four years?
d) The manufacturer offers to replace a pack if both racquets break in less than a guarantee period. What period should be set to limit the probability of replacing a pack to 0.01?
Question 5:
The effect of exercise on the amount of lactic acid in the blood of sportsmen was examined in a study. Eight males were selected at random from those attending a week-long training camp. Blood lactate levels (micrograms/L) were measured on each volunteer before and after playing three games of racquetball, with results as shown in the following table:
|
Player
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
|
Before
|
13
|
20
|
17
|
13
|
13
|
11
|
15
|
16
|
|
After
|
28
|
37
|
40
|
35
|
30
|
20
|
33
|
29
|
a) Is this paired or unpaired data?
b) Calculate the summary statistics relevant to this problem
c) Compute a 95% confidence interval for the mean change in blood lactate level.
Show all your working.
d) Based on that confidence interval, is there sufficient evidence to conclude that the blood lactate level rises as a result of training?
e) The investigators want to estimate the mean change in blood lactate level to an accuracy of ± 1.0 micrograms per Litre with 95% confidence, using the same measurement process (with a known standard deviation of 4). How many
Question 6:
In a study on the health and development of a bird species, Yellow Robin, two different sites were compared: one in The Dandenong Ranges (East of Melbourne) and another in the drier area of the Grampians (in western Victoria).
Samples of mature birds were taken at each site: 10 in The Dandenong Ranges and only 8 in the Grampians. They were trapped safely, weighed and then returned to the wild. The data on these sites is given in the following table:
|
Mass of Yellow Robin in Dandenong Ranges (g) 10.3 11.4 10.9 12.0 10.0 11.9 12.2 12.3 11.7 12.0
|
|
Mass of Yellow Robin in Grampians (g) 12.2 12.1 13.1 11.9 12.1 12.0 12.9 11.4
|
Consider the difference between the masses of the birds in the two regions.
a) Is this a paired or unpaired situation?
b) Calculate the summary statistics relevant to this problem.
c) Test whether there is evidence of any difference between the mean mass of the birds in the two regions. State clearly the null and alternative hypotheses and show all steps in the hypothesis test. Use long-hand calculation and check your answer using the Excel/Data Analysis appropriate tests.