Question 1 
The data for this question are stored in the file A1Q1.xls. This file contains the dividend yield (as a percentage) for 150 companies registered on the Australian Stock Exchange for the year 2005. The sample has been divided into two halves. Column A records the dividend yield for the largest 75 companies in the sample (measured by the value of the shares they have issued) while column B records the dividend yield for the smallest 75 companies in the sample. 

(a) Use Excel to find the mean and standard deviation for each group of companies. 
(b) Use Chebyshev's theorem to find the minimum proportion of observations that will lie within 1.5 standard deviations of the mean. 

(c) For each group of companies, use Excel to count the number of observations that lie within 1.5 standard deviations of the mean. Convert those numbers to proportions. Are they much larger than what you found in (b)? [Hint: Begin by counting the number of observations greater than x ?1.5s and less than x -1.5s . 

(d) If the dividend yields are normally distributed (the histogram is bell shaped), approximately 87% of the observations lie within 1.5 standard deviations of the mean. Based on the information in part (c), do you think dividend yields could be normally distributed? 

(e) Suppose you choose a small company share and a large company share at random. For each share, use the data to estimate the probability that the yield is less than 2%. 

(f) From the information in (a) and (e), what is the advantage of investing in smaller companies? What is the advantage of investing in larger companies? 

QUESTION 2 
The scores on the final exam in a statistics course have approximately a bell-shaped distribution. The mean score was 63.5 points and the standard deviation was 7.3 points. Suppose Pat, one of the students, had a score that was 2 standard deviations above the mean. 

What was Pat's approximate score? What can you say about the proportion of students who scored higher than Pat? [You are expected to respond to this question without using Normal distributions tables (or Excel)] 

QUESTION 3 
"MagTek" electronics has developed a smart phone that does things that no other phone yet released into the market-place will do. The marketing department is planning to demonstrate this new phone to a group of potential customers, but is worried about some initial technical problems which have resulted in 0.1% of all phones malfunctioning. The marketing executive is planning on randomly selecting 100 phones for use in the demonstration but is worried because it is very important that every single one functions OK during the demonstration. The executive believes that whether or not any one phone malfunctions is independent of whether or not any other phone malfunctions and is convinced that the probability that any one phone will malfunction is definitely 0.001. Assuming the marketing executive randomly selects 100 phones for use in the demonstration: 

(a) What is the probability that no phones will malfunction? [If you use any probability distribution/s, you are required justify the requirements for particular distributions are satisfied] ) 

(b) What is the probability that at most one phone will malfunction?

(c) The executive has decided that unless the probability of there being nomal functions is greater than 90%, he will cancel the demonstration. Should he cancel the demonstration or not? Explain your answer.

QUESTION 4
A Nobel Laureate, hosting a lecture for a large audience, is fed up with people who fail to turn their mobile phones off during such events. Based on numerous past performances he knows that the number of phones receiving calls during the lecture is normally distributed with a mean of 2.5 and a variance of 0.25. Before going onstage he tells his associate that if he hears more than 4 phone calls during tonight's lecture he will stop lecturing forever. 

(a) What is the probability that tonight's lecture will be his last? [Your answer should demonstrate your understanding of the distribution theory underpinning this question - i.e. avoid merely presenting a final figure based solely on an excel calculation]

(b) Assume you only knew the average number of phone calls received during the lecture is 2.5. (You did not know the variance and did not know if the number calls received during lecture is normally distributed). Use another distribution that you learnt to calculate the same probability as in(a).