Exercises page for Biostats Course VHM 801 at AVC - Fall Semester 2026

Follow this link to datasets (in Minitab (.mtw) and comma-separated (.csv) file formats).
Follow this link to extra exercises (labeled as x:number) and this link to AI discussion exercises.

This page contains links to solutions (either as text files, to be opened directly in a web browser or in Notepad (or similar), or as .pdf files, to be opened in a suitable reader, such as Adobe Acrobat) for selected exercises of VHM 801. The solutions have been compiled by the Biostats 801 course instructor, Henrik Stryhn.

Solutions to exercises

1. 1, 10, 16, 22, 42, 51, 65, 77, 110, 111, 113, 117, 121, 123, 127, 144, 145,
2. 7, 67, 69,
3. 4, 10, 14, 18, 19, 40, 52, 77, 94, 95,
4. 9, 10, 14, 26, 48, 52, 56, 60, 71, 73, 75, 76, 78, 92, 107, 108, 115, 122, 123,
5. 33, 47, 49, 51, 53,

Stata do-files for exercises

1. 10, 42, 51, 72, 77, 110, 111, 127, 145,
2. 7,
3. 14, 40,
4. 10, 71,
5. 49,

R program files for exercises

1. 10, 42, 51, 72, 77, 110, 111, 127, 145,
3. 14, 40,
4. 10, 71,
5. 49,

Data files

Data files (.mtw)

• Chapter 1: 10, 16, 23, 42, 51, 145,
• Chapter 2: 7,
• Chapter 3: 14, 40,
• Chapter 4: 48, 71, 75,
• Extra: parasite, PEI migration, crab, poplar, home assignment 2001:1,

Data files (.csv)

• Chapter 1: 10, 16, 23, 42, 51, 145,
• Chapter 2: 7,
• Chapter 3: 14, 40,
• Chapter 4: 48, 71, 75,
• Extra: parasite, PEI migration, crab, home assignment 2001:1,

Extra exercises

• x:1. This exercise uses the Mean and Median applet (use link from homepage) which allows you to place observations on a line and see their mean and median displayed visually. Follow the instructions to the left of the display. Put some points to get a sense of how it works.
  1. Now put two observations on the line. Why does only one arrow appear?
  2. Try next with three observations, where two are close together near the center and one somewhat to the right of these two. Move the rightmost point away and towards the other points, and observe how the mean and median behave. Explain your findings. Then move the rightmost across the two centre points to the left: what happened to the mean and median as you did that? Again, explain your findings.
  3. Finally, put five (distinct) observations on the line. Try to add one point without changing the median: where is your new point? Explore what happens to the median when you add another point; explain your findings.
  4. (Optional) If you have interest in exploring further, you may also try some of the suggestions below the display.

• x:2. In this exercise, we use the guinea pig data from Exercise 1.51 to explore how the number of bins in a histogram affects its shape and the impression it gives of the distribution.
  1. Draw first the default histogram (in your software). How many bins does it have?
  2. In Minitab, in order to change the binning of a histogram, you have to right-click on the bars, select the menu item Edit bars and the submenu Binning. Try to construct histograms with the maximal and minimal number of bins. How many bins did you get, and how does it affect your impression of the distribution?
  3. Experiment further with the binning to determine your preferred number of bins; how many bins did you choose? Does your preferred choice agree with the software default or any of the guidelines mentioned in Lecture 1? Do you think the shape of the distribution will affect the "best" number of bins? (For example, you may think about whether we should use the same number of bins for a symmetrical and a skewed distribution.)

• x:3. Randomization may be based on tables of random digits (e.g., Table A of PSLS or Table B of IPS). Discuss whether each of the following statements about such tables are true.
  1. Among 40 random digits (corresponding to one row in the table), there are exactly 4 nines.
  2. Each pair of digits has a chance/probability of 1/100 of being 99.
  3. The number 9999 can never appear as a group because it is not a random pattern.

• x:4. ABO blood types are determined from a pair of blood type alleles, inherited independently from the parents. Three alleles exist: A, B, and O, corresponding to the 4 blood types (with allele combinations indicated in parenthesis): A (AA and AO), B (BB and BO), AB (AB) and O (OO). Each allele may be inherited from a parent to a child with probability 0.5. Use this information to answer the following questions.
  1. Determine the blood types children of parents that both have blood type AB can have, and their probabilities.
  2. If the parents have allele combinations AB and AO, and they have two children, what is the probability that the children both have blood type A? What is the probability the two children have the same blood type?
  3. Which of the probabilities computed above are actually from binomial distributions?

• x:5. The Simulating the probability... and Coin Flip Simulation applets (link at homepage) allows us to simulate random tosses of a coin. We can use the applets to explore the behaviour of both relative frequencies (or proportions) and the counts of heads as the number of trials increases. Experiment a bit with the applets before focusing on the questions below. The Coin Flip Simulation applet will be easiest to use for the first questions of exercise. Set the number of coints at 1, p at 0.50 and use 10 Flips repeatedly.
  1. Record the number of heads after 50 tosses (flips), and compute the difference between the observed and expected proportion, as well as the difference between the observed and expected count. Hint: it may be easiest to enter the number of heads and the number of tosses in Minitab, and do the calculation in the Calculate menu (if you use formulas, the calculations will automatically expand to new rows).
  2. Continue the tosses up to totals of first 100, then 200 and 400.
  3. For 1000 tosses and more, the Simulating the probability... applet is easier to use because it allows 1000 tosses in one go. Perform trials with totals of 1000 and 2000, and repeat also here the calculations with the new numbers. You may continue with further tosses if you think it will be helpful.
  4. What patterns do you observe for the proportion and count of heads? Are these findings what you would expect?

• x:6. The Monty Hall problem: a famous statistical paradox that has generated a heated debate among statisticians and laymen. The following description is taken from the detailed Wikipedia article on the problem. The context is a game (television) show, described as follows:
  
  Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
  In order to properly understand the situation, it is important to explicitly define the role of the host:
  1. the host must always open a door that was not picked by the contestant,
  2. the host must always open a door to reveal a goat and never the car,
  3. the host must always offer the chance to switch between the originally chosen door and the remaining closed door,
  With these assumptions, determine the chance of winning the car in both of the options available to the contestant: (i) stay at the originally chosen door, and (ii) switch doors.

• x:7. The Graph-Probability Distribution Plot menu in Minitab computes and visualizes areas under the curve for many continuous distributions. In this exercise, we will use it to illustrate the 68-95-99.7 rule for the normal distribution.
  1. For a standard normal distribution, use the menu to calculate and visualize the area including values within one standard deviation on either side of the mean. Compare the area under the curve with what you would expect from the 68-95-99.7 rule.
  2. Repeat for areas including values within two and three standard deviations from the mean, and also here compare with the 68-95-99.7 rule.
  3. As discussed in the lecture, the length of human pregnancies may be approximated by a normal distribution with mean 266 and standard deviation 16 days. Use the 68-95-99.7 rule and/or the Minitab Probability Distribution Plot menu to answer the following questions.
     1. Almost all (99.7%) human pregnancies fall in what range of lengths?
     2. What percent of human pregnancies are longer than 282 days?

• x:8. We can use simulated data to explore how random variability affects statistical procedures. It is recommended to repeat each of the questions below a few times with different simulated data to get a sense of whether your findings are consistent or just reflect random noise.
  1. Generate 100 observations from the standard normal distribution. Produce a histogram with an overlaid normal distribution curve and a normal plot. Do these plots suggest any important deviations from a normal distribution?
  2. Generate 100 observations from the uniform distribution (0,1). Produce a histogram with an overlaid normal distribution curve and a normal plot. Do these plots suggest any important deviations from a normal distribution? Describe and explain any such deviations you see.

• x:9. Binomial versus hypergeometric distribution. We plan to carry out a survey among graduate students in one faculty of a small university, where the total population encompasses (only) 100 students. Our question of interest is whether the students own electronics (computer, tablet, phone) of a particular brand A. We plan to include 25 students in the survey. Assume that the true proportion in the population is 0.7 (or 70%) of owners.
  1. If the survey (quite unrealistically) was carried out with replacement (i.e., students could be selected to participate multiple times), what is the distribution of the number (X) of respondents saying they own electronics of brand A? Use statistical software to produce a tabulation of the probability function for the distribution of X, and produce also a graph of this distribution. What are the mean and variance of X (or X's distribution)?
  2. If the survey was carried out without replacement, the variable X would have a hypergeometric distribution with parameters 100 (population size), 25 (sample size) and 0.7 (true proportion). Use statistical software to produce a tabulation of the probability distribution of X and a graph of its distribution. Answer the following questions:
     1. Based on the graphs from a) and b), what is the visually biggest difference between the two distributions?
     2. Which value of X has the biggest difference in probability in the two distributions?
     3. Use your tabulated probabilities of the distribution in b) to compute its mean - do the two distributions have the same mean?
     4. Use your tabulated probabilities of the distribution in b) to compute its variance - do the two distributions have the same variance?
  3. How large should the survey be in order for it to be acceptable to use the same distribution for settings a) and b)?

• 2001 home assignment 1. link

Solutions to extra exercises:

Topics for discussion with a Generative AI bot

#	Topic	Suggested/Template questions
1	Choice of descriptive statistic	when to use a mean and when to use a median? (follow-up: if computing the median for ordinal categorical data, it may produce a nonsensical value?); what is a good descriptive statistic for spread of a distribution?; is the standard deviation appropriate for skewed distributions?
2	Outliers	what is an outlier?; are observations indicated by asterisks in a boxplot outliers? (follow-up: what do you mean, are they potential or actual outliers?); should outliers be removed from the data?
3	Association vs causation	can I conclude a causal effect from an observed effect?; can an experiment prove causation?; does association in a randomized controlled trial imply causation?
4	Probability (conditional)	A father of two children tells you one of his children is a boy. What is the probability the other child is a girl?; Suppose he then tells you the oldest child is a boy. What is the probability the younger child is a girl? (should these probabilities not be the same?)
5	Assessing normality	how to decide whether my data are normally distributed?; why not just use a normality test and go by p<0.05?; how to determine that a normal plot is straight enough?; which is the best normality test?