Suppose that Tom and Mary agree to meet for lunch at a
certain restaurant between
(a) First, without doing any calculations, make a prediction for the value of this probability.
(b) What do you think this probability might be if they agreed to wait thirty minutes for each other?
(c) Use the computer
to simulate one day’s arrival time for Tom and for Mary. If you measure the
arrival time in minutes after
MTB> rand 1 c1 c2;
SUBC> unif 0 60.
MTB> name c1 ‘Tom’ c2 ‘Mary’
(d) Record their arrival times, and indicate whether they arrive within fifteen minutes of each other and therefore have a successful meeting.
Tom’s arrival: Mary’s arrival: successfully meet?
(e) Now use the computer to simulate a large number, say 1000, of arrival times for Tom and for Mary. If you are using Minitab, the commands are:
MTB> rand 1000 c1 c2;
SUBC> unif 0 60.
(f) Examine visual displays (Graph>Histogram, Graph>Dotplot) of each person’s arrival time distribution separately. Do they appear to be fairly uniform over the (0,60) interval?
(g) Look at a scatterplot (Graph>Plot) of Mary’s arrival time vs. Tom’s. Do you see any pattern, or do the points appear to be randomly scattered?
(h) Create two new variables: (1) Tom’s arrival time minus Mary’s, and (2) the absolute value of this difference. In Minitab the relevant commands are:
MTB> let c3=c1-c2
MTB> let c4=abs(c3)
MTB> name c3 ‘Tom-Mary’ c4 ‘absdiff’
(i) Examine visual displays of the distributions of these two variables. Draw a rough sketch of them below, and write a sentence or two describing them.
(j) Use the computer to count how many of these 1000 simulated time differences are less than 15 minutes. What proportion of the 1000 simulated cases is this? In Minitab, you can use:
MTB> let c5=(c4<15)
MTB> name c5 ‘meet?’
MTB> tally c5
(k) Based on your simulation, does it seem to be more likely than not that Tom and Mary will meet? Explain.
(l) Re-create the scatterplot above using different labels for cases in which Tom and Mary meet (using the 15-minute waiting rule) as opposed to cases where they do not. You can do this in Minitab by using commands:
MTB> plot c1*c2;
SUBC> symbol c5.
(m) Use the same simulation results to approximate the probability that they would meet if they agree to wait 30 minutes for each other.
(n) Draw and label a square to represent Tom and Mary’s arrival times as ordered pairs, with Mary’s on the vertical axis. Then shade in the region of the square corresponding to their having a successful meeting. [You may want to start with the “x=y” line where they arrive at the same moment and think about how far from that line they can arrive and still meet. You may also want to use your simulation scatterplot to help guide your thinking.]
(o) Determine the area of this region as a fraction of the total area of the square. [Hint: It may be easier to find the area of the unshaded region first.] Because we are assuming that their arrival times are uniform and independent, this area is the probability of a successful meeting.
(p) Is this probability close to the approximate value you found with your simulation?
(q) Repeat this analysis to find the probability that they meet if they agree to wait thirty minutes for each other. Is this close to what your simulation suggested?
(r) Now let the number of minutes that they agree to wait be a variable, call it m. Follow your analysis above to express the probability that Tom and Mary meet as a function of m.
(s) Use the computer to graph this function for values of m ranging from 0 to 60. Draw a rough sketch of the function below, and write a sentence or two describing the behavior of this function. You might want to use Excel to graph this function, or in Minitab you can use:
MTB> set c6
DATA> 0:60
DATA> end
MTB> let c7=(insert function here)
(t) Determine how long each person would have to agree to wait in order for this probability to equal .5. [Find a solution algebraically, and verify it visually from the graph.]
(u) Determine how long each person would have to agree to wait in order for this probability to equal .9.
Now suppose that Tom and Mary’s arrival times are not
uniformly distributed between
(v) Would you predict that the probability of their meeting is higher, lower, or the same as before? Explain your thinking.
(w) Conduct a simulation analysis to approximate their probability of meeting. [To obtain random observations from this distribution, the Minitab subcommand is: normal 30 10.] As part of your analysis:
· Examine visual displays of the distribution of their individual arrival times to confirm that they appear to be normal with the given parameter values.
· Examine (and comment on) a scatterplot of Mary’s vs. Tom’s arrival time.
· Examine (and comment on) the distribution of the absolute value of the difference in their arrival times.
· Calculate the proportion of simulated samples in which they arrive within 15 minutes of each other and also within 30 minutes of each other.
· Write a report describing your findings, comparing them to the results with uniform distributions, and explaining why it makes intuitive sense for the results to differ in the way that they do.
(x) Use the result that linear combinations of normal random variables are themselves normally distributed to calculate the probability of Tom and Mary arriving within fifteen minutes, and then within thirty minutes, of each other. Compare these theoretical answers to your simulation results. [Hint: Consider the probability distribution of the difference in their arrival times.]
(y) Continue to assume that Mary’s and Tom’s arrival times are independent and normally distributed.
·
Describe how you would expect the probability of their meeting to
change if the standard deviations of their arrival times were each 5 minutes
rather than 10 (with both means at 30 minutes past
· Describe how you would expect the probability of their meeting to change if Mary’s arrival time had a mean of 20 minutes past noon and Tom’s had a mean of 40 minutes past noon (with both standard deviations equal to 5).
· Conduct simulation analyses to approximate the probability that they meet, using both the 15- and 30-minute rules, for these two situations.
· Use the bivariate normal distribution to calculate the exact probabilities of their meeting in these two cases, and compare these probabilities to your simulation results.
· Write a paragraph or two describing your findings, comparing them to your predictions in a) and b), and explaining why they make sense.
(z) Continue to suppose that Tom and Mary’s arrival times follow a normal distribution with mean 30 minutes and standard deviation 10 minutes, but relax the assumption that their arrival times are independent. Use both a simulation analysis and a theoretical analysis to investigate the probability that they meet (according to the 15-minute rule) if the correlation coefficient between their arrival times is:
· 0.2
· 0.6
· –0.4
· Comment on the effect of the correlation coefficient on their probability of meeting.
· Explain what the correlation coefficient reveals about Tom and Mary’s arrival times in this situation. Do you think that this correlation would likely be positive?