Data Analytics Module
Lecturer: Hans van der Zwan
Lab 03
Topic: modelling


EXERCISE 3.1

A multiple choice exam consists of 50 questions with four alternatives each. A student makes the exam by filling in the answers at random. Calculate the probabilities for filling in 0, 1, 2, …., 50 correct answers. Plot these probabilities in a bar graph.


EXERCISE 3.2

Milk bottles are machine-filled with a mean of 1000 ml and a standard deviation of 10 ml. Assume the content of the bottles has a normal distribution.

  1. Calculate the percentage of bottles that contain less than 990 ml.
  2. Calculate the percentage of bottles that contain less than 1010 ml.
  3. Calculate the percentage of bottles that contain more than 1010 ml.
  4. Calculate the percentage of bottles that contain more than 985 ml.
  5. Calculate the percentage of bottles that contain between 990 and 1010 ml.


EXERCISE 3.3

According to an EU directive the weights of packs of sugar should have a mean of 1000 gram with a maximum negative tolerable error of 7.5 gram (76/211/EEC).
Assume that the weights of packs of sugar are normally distributed with mean 1000 and standard deviation 5 gram.

  1. What percentage of the packs of sugar has a weight beneath what is allowed according to the EU directive, i.e. beneath 992.5 gram?
  2. Assume the mean weight is set at 1005 gram. What is the effect on the percentage that does not comply with the EU directive?
  3. The mean weight is 1005 gram. What should be the value of the standard deviation to assure that at most 0.5% of the packages does not comply with the EU directive?


EXERCISE 3.4

Normal distributions play an important role in inferential statistics, i.e. the part of the field in which sample results are projected onto the population.
In many cases z-values are used and especially the lower and upper tail probability of z-values (z-scores).
E.g. in a normal distribution the upper tail probability of a z-score of 2 is about 0.0225.

  1. What is the upper tail probability if the z-score = 1.5?
  2. What is the upper tail probability if the z-score = 1.65?
  3. What is the lower tail probability if the z-score = -1.90?
  4. What is the lower tail probability if the z-score = -1.28?
  5. What is the lower tail probability if the z-score = 1.58?
  6. Which z-score has an upper tail probability of 0.10?
  7. Which z-score has a lower tail probability of 0.05?