Question 1  [10 Marks]

In lectures we discussed a case study in which researchers investigated whether hair grows back thicker after  shaving (see http://www.scientificamerican.com/article/fact-or-fiction-if-you-shave-or-wax-your-hair-will-come-back-thicker/).

1. Design a study to investigate this question using our STA1010 class as a sample. Assume the class has 200 students: 100 male and 100 female. Describe  the experiment and illustrate using a schematic diagram. Treat males and females separately.
2. What is the population being sampled? What type of sampling is this? Discuss whether the sample is representative of this population.
3. Comment on aspects of control, randomisation, replication and blocking.
4. Say what comparisons you would make at the end of the experiment.

Question  2  [8 Marks]

A researcher wants to know whether lack of sleep affects the scores soldiers achieve in a test of shooting ability. Ten volunteers slept the night before taking the test (S). The other nine were prevented from sleeping for 48 hours prior to the test (NS).

Volunteer	1	2	3	4	5	6	7	8	9	10
Sleep (S) or No Sleep (NS)	S	NS	NS	S	NS	S	NS	NS	NS	S
Test score	25	20	23	14	26	29	20	10	18	22

Volunteer	11	12	13	14	15	16	17	18	19
Sleep (S) or No Sleep (NS)	S	NS	NS	S	S	NS	S	S	S
Test score	20	22	24	18	19	10	28	23	20

• State the five number summariesof the measurements for the two data sets.
• Check for outliers in each data set and state the value of any outliers found.
• Construct a comparative box-plot for the two data sets.
• Scores are scaled by doubling them and adding 40. What are the new medians and interquartile ranges for each group?

Question 3  [6 Marks]

A study was made in a recent year of the number (in thousands) of university degrees awarded in different general discipline areas across three geographical areas.    The numbers in each category were as follows:

	Engineering	Natural Sciences	Social Sciences
U S A	62	111	182
Western Europe.	159	140	116
Asia	281	243	236

1. Find the overall proportions of degrees awarded (a marginal distribution) in each of the three discipline areas.
2. Find the conditional probabilities (distribution) of a student having graduated in each of the three discipline areas, given that the student is from Asia.
3. What is the probability that a graduate randomly chosen from the USA has a degree in engineering?
4. Do the data suggest Discipline Area and Geographical area are dependent? Explain your answer.

Question 4 [9 Marks]

A test for Crohn’s disease classifies 95% of people with the disease as affectedand 4% of those who don’t have the disease as affected. It is known that about 0.05% of the population has Crohn’s disease.

1. What are the false positive and false negative rates?
2. What is the probability that someone classified as affected does in fact have the disease?
   1. Solve this problem by drawing up a contingency table, AND
   2. Solve using conditional probability and the law of total probability.

Question 5  [10 Marks]

A clown is fired out of a cannon at an initial velocity of u metres per second. The cannon makes an angle qto the ground (see diagram).

The horizontal distance R travelled by the clown should in principle obey the following formula:

where g is 9.8 m/s² and q is measured in radians.