• The number of marks available for each question is given in parenthesis. There is a total of 50 marks.
  • Please submit this electronically via email. You may either type this up, or scan in handwritten solutions.

Question A (9)

You are given one 6-sided die and one 8-sided die. You roll them together and obtain a score \(X\) by summing the outcomes of both dice.

  1. Calculate the pmf of \(X\).
  2. Calculate the cdf of \(X\).
  3. Calculate \(\mathbb{E}[X]\).
  4. Calculate \(Var(X)\).
  5. Calculate the median of \(X\).

Question B (7)

A rare blood disease is known to affect around 1 in every 2000 children. A test for the disease is 95% accurate. A child is selected at random and is tested for the disease. What is the probability that the child actually has the disease?

(Hint: you should probably be using Bayes’ theorem somehow.)

Question C (7)

  1. Explain the differences between nominal, ordinal, interval, and ratio data.
  2. Give an example of each of the four types of data: nominal, ordinal, interval, and ratio data.

Question D (12)

You each have a 7 or 8 digit student number. Subtract each digit from 9, then, by treating each results as a piece of ratio data (e.g. if your student number is 1016865, then you have the data set \(\{8, 9, 8, 3, 1, 3, 4\}\) with seven or eight values), calculate its:

  1. mode,
  2. median,
  3. mean,
  4. range,
  5. IQR,
  6. variance,
  7. standard deviation.
  8. Draw the boxplot of the data.
  9. Comment on its skewness.

Question E (8)

For each of the scenarios below, give an appropriate statistical test, stating the null and alternative hypotheses:

  1. A botanist would like to know if lemon trees produce more fruit when planted in soil rich in zinc or in soil deficient of zinc. They plant 30 lemon trees in pots with zinc rich soil, and 40 lemon trees in pots with zinc deficient soil. After 2 years the number of lemons each tree produces are counted.

  2. The Welsh government would like to know if there people living in urban areas are more likely to speak a foreign language than people in rural areas. They randomly choose 240 participants and ask them if they live in a rural or urban area, and if they speak a foreign language or not.

  3. A disreputable brewing company is claiming to sell low alcoholic beer with only 0.5% alcohol. However there are reports that the public are getting quite tipsy from drinking the beverage. An inspector, keen to discover the truth, takes a sample of 200 bottles of the beer and measures their alcohol content.

  4. The Tokyo Olympic committee find a new flavour of Coca-Cola is incredibly popular amongst track and field athletes. The Coca-Cola company claim that the drink “definitely, in no way, makes you run faster”, but the committee is suspicious. To determine weather the drink should be banned at the next Olympics in Paris, an experiment is conducted: 30 athletes record their 100m sprinting times before and drinking the beverage.

Question F (4)

You are keen to conduct a survey amongst your classmates to see if they have enjoyed the MA0004 module.

Identify two types of sampling bias that might arise. Explain how and why they might arise. Give a suggestion on how to avoid these biases.

Question G (3)

Investigate a phenomenon known as collider bias. Explain the phenomenon briefly, and give a simple example.