STAT1122 Biostatistics Assessment

Question 1 – Probability [Total: 15 marks] 

A gumball machine located near David Jones at the Marion Shopping Centre in Adelaide contains 1568 gumballs. It is filled monthly with America's original Dubble Bubble Gumballs. Each gumball is approximately 2cm in diameter. The original gumballs have a solid colour coating and come in 8 flavours; strawberry, pineapple, blueberry, lemon-lime, orange, cherry, banana and grape. However, the machine also contains Berry Blast Speckled Dubble Bubble Gumballs (and as the name suggests have a speckled or mottled coating).

The colours/flavours/coatings in the machine in this particular week are: strawberry (solid red coating), Blueberry (solid blue coating), banana (solid yellow coating), orange (solid orange coating), strawberry (speckled red coating), lemon (speckled yellow coating) and blueberry (speckled blue coating). Six hundred and ninety are speckled and 36.224% are yellow. Of the red ones, 198 have a solid colour coating and 201 are speckled. There are 344 blue solid coated gumballs. Only 3.316% are banana flavoured.

A child inserts a coin into the gumball machine to obtain a single gumball at random. What is the probability of the following events?

Hint: A cross tabulation or contingency table can help. 

1. What is the probability that the child randomly selects a gumball that is strawberry flavoured and is speckled? [5 marks]
2. What is the probability that the child randomly selects a gumball with a solid colour coating or one that is flavoured as blueberry? [5 marks]

• What is the probability that the blueberry flavoured gumball chosen by the child at random turns out to be speckled? [5 marks]

1. What is the probability that the child randomly obtains a strawberry flavoured gumball? [5 marks]
2. What is the probability that the child randomly selects a banana flavoured gumball? [5 marks]
3. What is the probability that the child randomly selects a gumball that is blueberry flavoured given that it is speckled? [5 marks]

• What is the probability that the child randomly obtains a speckled gumball? [5 marks]
• Are the events “blueberry flavoured” and “speckled” independent? Demonstrate this Then provide a brief description of what independence refers to in this specific case. [4.5 marks] 

Question 2 – Binomial probability models [Total: 10 marks]

1. What are the 4 key assumptions that must be satisfied in order to correctly use the binomial probability distribution to answer the questions below. Give examples for each using the scenarios below to justify your answer. [2 marks] 

Scenario 1: A young driver is about to sit a 25-question multiple choice test for their driver’s licence. They did not study for this particular test and hope to randomly guess the correct answer for each question without considering which choices would be more correct than others. They are quite hopeful given that there are only 3 choices to choose from for each question and that only one choice is exactly correct. 

1. If this particular young driver blindly guesses the answer to each question, what is the probability that they will get only 15 questions correct? Let X be the total number of correct X has a binomial distribution: ~Bin(n,p) [2 marks] 

• If this particular young driver blindly guesses the answer to each question, what is the probability that they will get at least 19 questions correct? Let X be the total number of correct X has a binomial distribution: ~Bin(n,p) [2 marks] 

Scenario 2: A pet female bunny (doe) has just delivered a litter of 8 kits. The sex of each kit is random and equally likely. Each kit’s sex is independent of the previous and of the next. 

1. What is the probability that the doe delivers more than 6 female kits? Let Y be the total number of female kits. Y has a binomial distribution: ~Bin(n,p) [2 marks]
2.  What is the probability that the doe’s litter is half male? Let Y be the total number of male Y has a binomial distribution: ~Bin(n,p) [2 marks]

Question 3 – Normal probability models [Total: 10 marks]

 You want to go to Mt Hotham for snowboarding sometime in the July- August season (defined as the days inclusive of July-August only), as you can see that the peak in “days with significant snowfall” occurs across these 2 months. 

The number of ‘predominant snowfall’ days (defined when snowfall is ? 5 cm) in the days of July-August varies according to a Normal distribution with a mean of 30 days of snowfall across that particular period and standard deviation of 5 days. 

1. If you were to choose a year at random, what is the probability that the year you choose has less than 15 days of predominant snowfall across July-August period? [5 marks] 
2. What is the proportion of years where July-August’s snowfall falls between 10 and 20 days? [5 marks] 
3. What is the proportion of years where July-August’s snowfall is greater than 40 days? [5 marks]
4.  Considering the 50% snowiest July-August seasons, up to what number of days of predominant snow (? 5 cm) would these snowiest July-August seasons receive? [5 marks]

 Question 4 – Statistical Inference: Hypothesis testing and estimation [Total: 15 marks]

 A recent survey of 3098 employees of a global HR company were asked at random whether they were satisfied or unsatisfied with the office equipment that the company supplied. Two thousand and three employees reported to be satisfied with the supplied office equipment. 

1. Construct a 90% confidence interval for the (population) proportion of employees surveyed and felt satisfied with their office equipment. What does this indicate? Write an appropriate contextual [4 marks] 
2. It has previously been reported that the typical rate of employee satisfaction of office equipment is 3:1, that is for every 3 employees that are satisfied there is 1 employee that is not satisfied. Test the claim that the rate of employee satisfaction significantly differs from this ratio. Use all steps for hypothesis testing to make a conclusion at a significance level (alpha) of 5%. [9 marks]

HINT: Write an appropriate contextual statement at STEP 6. 

• What assumptions or conditions are made when conducting these statistical inference tests above? Are they being satisfied? If so or if not, you must demonstrate these [2 marks]