Question 1.

A group of students was interested in testing a claim that a mechanical citrus juicer will get more juice from fruit than hand squeezing the fruit. They purchased a range of oranges from a supermarket. The oranges were randomly divided into two groups - one group was juiced by hand, the other group was juiced using the mechanical citrus juicer.

The data for this comparison is stored in the file "Orange Juice.csv" which can be downloaded from Canvas. The data contains 2 variables:

Volume The volume of juice obtained from the orange (in millilitres)
Method The method used to juice the orange (Hand, Juicer).

(a) Briefly explain why this study is an experiment.

(b) (i) Run the VIT software and load the file Orange Juice.csv into it. Run a randomisation test to compare the median volume of juice obtained by the two methods. Include the output in your assignment answers.

(ii) When chance is acting alone, would it be unusual to get a difference between the two group medians at least as big as the observed difference?

(iii) Is it plausible that the observed difference between the two group medians can be explained by "chance acting alone"? Briefly justify your answer.

(iv) Can we conclude that using the mechanical citrus juicer results in a higher median volume of juice than if juicing the oranges by hand? If so, justify why with two reasons. If not, what can we conclude?

Question 2.

A banker is interested in assessing the performance of staff at a particular branch. She is considering 3 different methods of collecting data on this.

Method 1: For two weeks, have signs in the branch requesting that customers fill in survey forms available from the counter by the tellers and then place the completed forms in a box by the entrance.

Method 2: For two weeks, each employee would ask every 10th customer they deal with if they were willing to fill in a survey. If so, give them a survey form to complete and ask them to post the completed form in a box by the entrance.

Method 3: Use the bank's records to take a random sample of 500 people with accounts at the branch and post a questionnaire to those selected asking them to fill in and return it. People who returned the survey would go into a draw to have $100 deposited into their account.

(a) For which of the methods, if any, is non-response bias a potential problem?
(b) For which of the methods, if any, is self-selection bias a potential problem?

(c) Which of the methods, if any, is subject to sampling errors?

(d) (i) Give an example of why there may be selection bias using Method 1.

(ii) Give an example of why there may be selection bias using Method 3.

(e) What type of bias would the offer of the prize draw in Method 3 be trying to minimise?

(f) Describe another potential source of bias with method 1 that has not already been discussed in parts (a) to (e) above

Question 3.

A wildlife biologist was interested in whether raising deer in captivity has an effect on the size of the deer. She took a random sample of one-year old deer that had been raised in the wild and another sample of one-year old deer of the same breed that had been raised on a deer farm and obtained the animals weights.

The data are stored in the file "Deer.csv" which can be downloaded from Canvas. The data contains 2 variables:

Weight The weight of the deer (in kilograms)
Environment The environment the deer was raised in (Wild, Farm)

Run the VIT software and load the file Deer.csv into it.

(a) (i) Generate a bootstrap confidence interval for the mean weight of one-year old deer. (DO NOT use the variable Environment at this point.) Include the output in your assignment answers.

(ii) What is the parameter we are estimating using this bootstrap confidence interval?

(iii) Do we know the true value of this parameter?

(iv) Interpret the bootstrap confidence interval.

(v) Briefly explain why students doing this assignment will not all get the same bootstrap confidence interval.

(b) (i) Generate a bootstrap confidence interval for the difference between the mean weight of one-year old deer that have been raised in the wild and one-year old deer that have been raised on a farm. Include the output in your assignment answers.

(ii) What is the parameter we are estimating using this bootstrap confidence interval?

(iii) Interpret the bootstrap confidence interval.

(iv) Based on the bootstrap confidence interval, is it believable that the mean weight of one-year old deer that have been raised in the wild is the same as that of one-year

Question 4.

A statistics student was interested in investigating how long it takes to get a pizza delivered from the local pizzeria. Over a few weeks, a random sample of 10 delivery times (in minutes) was recorded. The data are displayed below:

16.3, 29.7, 18.5, 27.4, 19.8, 23.8, 23.4, 19.2, 28.4, 21.3
Summary Statistics: x = 22.78 minutes. And s = 4.55

(a) Calculate and interpret a 95% confidence interval for the mean delivery time.

Note: You must clearly show that you have followed the "step-by-step guide to producing a confidence interval by hand" given in the Lecture Workbook, Chapter 6. Use the t-procedures tool to find values for t-multipliers and standard errors.

(b) A friend of the statistics student pointed out to them that their confidence interval must be wrong as "more than half the values used to calculate it were not in the 95% interval, so how can they have 95% confidence in the interval?" Explain what was wrong with this thinking.

Question 5.

In 2010, PopCap Games commissioned a survey of adult mobile phone gamers in the United States and in the United Kingdom. A similar survey was also carried out in 2009. Some of the information from the survey is given below:
Do you play mobile phone games for less than 60 minutes per week?
2009: Yes (521) Total
(659)
2010: Yes (554) Total (814)
How often do you typically play on your mobile phone?

464 of the gamers surveyed in 2010 had recommended games to others. This group was asked what influenced them to recommend a game. They could choose as many answers as they wished. The answers, with the percentages, selecting them were:

33% From a trusted brand
38% Controls simple to use
49% Easy to learn
86% Fun to play
40% Good value
36% Great graphics and sound
45% Is challenging
41% Lots of levels/modes

(a) State the sampling situation (a, b or c) for calculating the standard error of the difference in the following scenarios:

(i) estimating the difference between the proportion of gamers that played for less than 60 minutes per week in 2009 and the proportion of gamers that played for less than 60 minutes per week in 2010.

(ii) estimating the difference between the proportion of gamers that played for at least 60 minutes per week in 2010 and the proportion of gamers that played daily in 2010.

(iii) estimating the difference between the proportion of gamers that played daily in 2010 and the proportion of gamers that played 2-3 times per week in 2010.

(c) For gamers who recommend games to others, calculate and interpret a 95% confidence interval for the difference between the proportion of all gamers who, in 2010, would have recommended a game that is fun to play and the proportion of all gamers who, in 2010, would have recommended a game that is easy to learn.