Name:
Murphy’s Law and Tumbling Toast
(Read this in an English accent.) If a piece of toast falls off your breakfast plate, is it more likely to land with the
buttered side down? According to Murphy’s Law (the assumption that if anything can go wrong, it will), the
answer is “Yes.” Most scientists would argue that by the law of probability, the toast is equally likely to land
butter-side up or butter-side down. Robert Matthews, science correspondent of the Sunday Telegraph,
disagrees. He claims that when toast falls off a plate that is being carried at a “typical height,” and the toast
has just enough time to rotate once (landing butter-side down) before it lands. To test his claim, Mr. Matthews
has arranged for 150,000 students in Great Britain to carry out the experiment with tumbling toast.
Assuming scientists are correct, the proportion of times that the toast will land butter-side down is p = 0.5. We
can use a coin toss to simulate the experiment. Let heads represent the toast landing butter-side up.
1. n = 5
a) Toss a coin 5 times and record the proportion of heads obtained, p̂  (number of heads)/n. Explain how
your result relates to the tumbling-toast experiment.
b) Repeat this sampling process 4 more times. Make a dotplot of the 5 values of p̂ and sketch it below. Is
the center of your distribution close to 0.5?
c) Five repetitions give a very crude approximation to the sampling distribution. Pool your results with the
rest of the class to obtain many repetitions. In a different color, add the class values above to make a
histogram of all the values of p̂ . Is the center close to 0.5? Is the shape approximately normal?
d)
How much sampling variability (standard deviation) is present? That is, how much do your values of
p̂ based on sample sizes of size 5 differ from the actual population proportion, p̂  0.5?
2. n = 15 Now repeat parts (a) to (d), but this time toss the coin 15 times.
a) Toss a coin 15 times and record the proportion of heads obtained, p̂  (number of heads)/n. Explain
how your result relates to the tumbling-toast experiment.
b) Repeat this sampling process 4 more times. Make a dotplot of the 5 values of p̂ . Is the center of your
distribution close to 0.5?
c) Again, four repetitions give a very crude approximation to the sampling distribution. In a different color,
pool your results with the rest of the class above to obtain many repetitions. Make a histogram of all the
values of p̂ . Is the center close to 0.5? Is the shape approximately normal?
d)
How much sampling variability (standard deviation) is present? That is, how much do your values of
p̂ based on sample sizes of size 15 differ from the actual population proportion, p̂  0.5?
3. n = 40
a) Toss a coin 40 times and record the proportion of heads obtained, p̂  (number of heads)/n. Explain
how your result relates to the tumbling-toast experiment.
b) Repeat this sampling process 4 more times. Make a dotplot of the 5 values of p̂ . Is the center of your
distribution close to 0.5?
c) Five repetitions give a very crude approximation to the sampling distribution. In a different color, pool
your results with the rest of the class above to obtain many repetitions. Make a histogram of all the
values of p̂ . Is the center close to 0.5? Is the shape approximately normal?
d)
How much sampling variability (standard deviation) is present? That is, how much do your values of
p̂ based on sample sizes of size 40 differ from the actual population proportion, p̂  0.5?
4. Why do you think Mr. Matthews is asking so many students to participate in his experiment?
5.
What is the largest possible sample you can take from this population and still be able to calculate the
standard deviation of sampling distribution of
using the method presented in the textbook?
6. What do you notice about the standard deviation as the sample size increases?
7. What happened in the experiment with the students and the toast? Watch this riveting physics
experiment to find out! https://www.youtube.com/watch?v=zwIfBSbNQCs
Homework Problem Set:
1) According to the 2000 U.S. Census, 80% of Americans over the age of 25 have earned a high school
diploma. Suppose we take a random sample of 120 Americans and record the proportion,
of
individuals in our sample that have a high school diploma.
(a) What are the mean and standard deviation of the sampling distribution of
(b) What is the approximate shape of the sampling distribution? Justify your answer.
(c) Suppose our sample size was 30 instead of 120. Compare the shape, center, and spread of this
sampling distribution to the one in parts (a) and (b).
(d) You live in a small town with only 500 residents over the age of 25. What is the largest possible sample
you can take from your town and still be able to calculate the standard deviation of sampling
distribution of
using the method presented in the textbook? Explain.
2) Suppose you are going to roll a fair six-sided die 60 times and record
2 is showing.
the proportion of times that a 1 or a
(a) What is the mean of the sampling distribution of
(b) What is the standard deviation of the sampling distribution of
(c) Describe the shape of the sampling distribution of
. Justify your answer.
(d) Suppose that when you actually roll the die 60 times, you get 30 rolls of 1 or 2, for a
suspicious about whether the die is fair? Justify your answer.
of 0.5. Are you
3) The superintendent of a large school district wants to know what proportion of middle school students in her
district are planning to attend a four-year college or university. Suppose that 80% of all middle school students
in her district are planning to attend a four-year college or university. What is the probability that a SRS of size
125 will give a result within 7 percentage points of the true value?
4) Are attitudes toward shopping changing? Sample surveys show that fewer people enjoy shopping than in
the past. A recent survey asked a nationwide random sample of 2500 adults if they agreed or disagreed with
the statement, “I like buying new clothes, but shopping is often frustrating and time-consuming.” In this survey,
1520 agreed. Suppose that in fact 60% of all adult U.S. residents would say “Agree” if asked the same question.
(a) What is the sample proportion of U.S. adults who agreed with the statement?
(b) If, in fact, the proportion of all U.S. adults who would agree with the statement is 0.60, what is the
probability that the proportion in a random sample of 2500 adults is as far from 0.60—above or below—
as the results of this survey? (Check that the necessary conditions are met before calculating this
probability.)