Question 1.

When drawing a rectangle or creating a rectangular object, a particular ratio of width to length is thought to be most aesthetically pleasing. This ratio is ( √5 - 1)/2 = 0.618, and is called the golden ratio. The Shoshoni Indians (native Americans) used beaded rectangles to decorate their leather goods. Do the beaded rectangles of the Shoshoni follow the golden ratio? A random sample of beads was measured. The data is stored in the text file "shoshoni", which contains the variable:

shoshoni the ratio of width to length for the rectangular bead

Question 2.

Twelve subjects were each tested twice to measure their reaction time to a given stimulus, once after taking a drug and once after taking a placebo. The order of the treatments was randomised. Does the drug have any affect on the reaction time? The data is stored in the text
file "reaction", which contains the variables:

drug      the time to react after being given the drug (in seconds)
placebo the time to react after being given the placebo (in seconds)

Question 3.

A sample of 21 exam marks was randomly selected from a large stage 2 course. The marks were: 4, 5, 6, 7, 8, 11, 12, 14, 15, 20, 21, 27, 30, 33, 35, 40, 41, 50, 65, 69, 98

(a) Enter the data into R as a data vector called y.

(b) Create a new data vector log.y by using a natural log transformation on y. List log.y.

(c) (i) What effect has the transformation had on the lowest exam mark?
(ii) What effect has the transformation had on the highest exam mark?
(iii) Does the log transformation affect the higher or lower values more?

(d) (i) Produce a vertical box plot for y.
(ii) Produce a vertical box plot for log.y.
(iii) Produce side-by-side box plots for y and log.y.

Hint: start by creating two new variables as follows:
y.data<-c(y,log.y)
y.label<-factor(rep(c("y","log.y"),c(21,21)))

(iv) Briefly comment on what these plots show.

(e) (i) Use the R function summaryStats to calculate the major summary statistics for both y and log.y.

(ii) Compare the mean of y with the median of y.

(iii) Compare the mean of log.y with the median of  log.y.

(iv) Comment on the skewness statistics for both y and log.y.

(v) Using R as a calculator, back-transform the mean of log.y.

(vi) Is the back-transformed mean of log.ycloser to the mean of y or to the median of y?

Question 4.

A study was conducted by two University of Auckland staff (a statistician and a recreation officer) to assess the impact of new rugby rules that were aimed to improve the flow of the game. A random sample of 100 time periods of uninterrupted play were recorded from games
played under the new rules and another random sample of 100 time periods of uninterrupted play were recorded from games played under the old rules. The data is stored in the text file "rugby2", which contains the variables:

times     the time recorded for a period of uninterrupted play (in seconds)
rules      the rules that were played under:
             new or old
The university staff were interested in whether the times of uninterrupted play differed under the two sets of rules, and, if so, by how much.

Question 5.

A university lecturer was interested in assessing the fuel economy of her new car. Each time she filled up with petrol, she noted the distance travelled and the quantity of fuel required to fill the tank. It was of particular interest to predict the distance trave lled on 40 litres of fuel. The data are stored in the text file "fue lconsume", which contains the variables:

litres   the amount of petrol used (in litres)
dist     the distance travelled (in km)