18C InClass

How well can humans distinguish one Martian letter from another? In this activity, we’ll find out. When shown two Martian letters, Kiki and Bumba, write down whether you think Bumba is on the left or on the right.

A photograph of Mars.

Credit: iStock/24K-Production

Question 1

1) Were you correct or incorrect in identifying Bumba?

Question 2

2) What are the observational unitsfor the data we just collected?
a)Martians
b)Letters
c)Students
d) Kiki and Bumba

Question 3

3) What variable did we collect on each observational unitin Question 1?
a)Whether a student w correct or incorrect in identifying Bumba
b)Whether Bumba is on the left or on the right
c)The number of students who were correct in identifying Bumba
d)Whether a student can read Martian

Question 4

4) Is the variable you identified in Question 3 categorical or quantitative?

Question 5

5) Now that we have collected data, we will summarize the data by calculating a statistic.
Part A: How many students are in class today? (This is your sample size.)
Part B: How many students in your class were correct in identifying Bumba?
Part C: Use the values from Parts A and B to calculate the proportion of students who correctly identified Bumba. This is the observed value of our summary statistic, a sample proportion.Write this value using proper statisticalnotation.
To determine if these data provide evidence that the class can read Martian, we will simulate what would happen in the class if we can’t read Martian, repeat the simulation many times to understand what results would be surprising if students were just guessing, and then compare the class’s observed data to the simulation. This gives us an estimate of how often (or the probability of) the class’s result would occur just by chance if students were all just guessing. If our observed data were unlikely to occur, the assumption of “randomly guessing” is not plausible, and we would interpret that as evidence that students were not just guessing.

Question 6

6) Assume that humans really don’t know Martian and are just guessing which letter is Bumba.
Part A: What is the probability of guessing Bumba correctly?
Part B: Describe how we could use a coin to simulate each student “just guessing” which Martian letter is Bumbaand whether the guess is correct.
Part C: How could we use a coin to simulate the entire class “just guessing” which Martian letter is Bumba?
Part D: How many people in your class would you expect to choose Bumba correctly just by chance? Explain.

Question 7

7) Each student will flip a coin one timeto simulate a“guess” under the assumption that we can’t read Martian. Let heads = correctandtails = incorrect.
Part A:What was the result of yourcoin flip?
Part B:What was the result from your class’s simulationof 𝑛student guesses, where 𝑛is your sample size from Question 5,Part A? What proportion of students “guessed” correctly in thissimulation?

Question 8

8) If students really don’t know Martian and are just guessing which is Bumba, which seems more unusual: the result from your class’s simulation in Question 7,Part B or the observed proportion of students in your class whowere correct in Question 5? Explain.

While your observed class data are likely far different from the simulated “just guessing” class, comparing your class data to a single simulation does not provide enough information. The differences seen could just be due to the randomness of that set of coin flips! Let’s simulate another class.

Question 9

9) Each student should flip their coin again.
Part A:What was the result from your class’s second simulationof 𝑛student guesses, where 𝑛is your sample size from Question 5,Part A? What proportion of students “guessed” correctly in thissecond simulation?
Part B:Create a dotplot to compare the two simulatedresults with the observed class result.
We still only have a couple of simulations to compare our class data to. It would be much better to be able to see how our class compared to hundreds or thousands of “just guessing” classes. Since we don’t want to flip coins all class period, your instructor will use a computer simulation[1] to get 1,000 trials.

Question 10

10) Fill in the following valuesto describe how we would create a simulation of random guessing with 1,000 trials (repetitions).Population Proportion(correct guess) = Sample Size= Number of Samples =

Question 11

11) Sketch the distribution displayed by your instructor here. Label thex-axisappropriately.

Question 12

12) Is your class particularly good or bad at reading Martian? Use the plot in Question 11to explain your answer.

Question 13

13) Is it possiblethat we could see our class results just by chance if everyone was just guessing? Explain.

Question 14

14) Is it likelythat we could see our class results just by chance if everyone was just guessing? Explain.

One way to quantify your answers to Questions 12–14 is to calculate the proportion of simulated samples in which the proportion of students who guessed Bumba correctly was equal to or larger than the one we observed in Question 5. This proportion is an estimate of the P-value—the probability of observing a sample proportion as or more extreme as ours, assuming the null hypothesis is true.

Question 15

15) Write out the null hypothesis in this study.

Question 16

16) Write out the alternative hypothesis in this study.

Question 17

17) Let 𝑝denote the true probability that a student will guess Bumba correctly. Write the null and alternative hypotheses in terms of the parameter 𝑝using appropriatestatistical notation.

Question 18

18) Use the distribution of simulated sample proportions to estimate the P-value.

Question 19

19) Use your answer to Question 18to determine ifthis activity providesstrong evidence that students were notjust guessing at random. Explain.

Question 20

20) If these data gave us evidence that students were not just guessing at random, what do you think is going on here? Can we as a class read Martian?[2]

  1. https://dcmathpathways.shinyapps.io/SampDist_prop/ ↵
  2. Ramachandran, V. (2007, March). 3 clues to understanding your brain[Video]. TED. http://www.ted.com/talks/vilayanur_ramachandran_on_your_mind ↵