In Exploration 1.1 you looked at data involving the dog Harley picking one of two cups that the experimenter bowed toward. Another dog in that study named Jango, a male black lab, went to the cup that the experimenter bowed toward 8 out of 10 times. Does Jango understand bowing? Let π represent Jango’s probability of going to the correct cup. We will test the hypotheses H₀: π = 0.50 and H_a: π > 0.50. 

a. Use the One Proportion applet to give a simulation-based p-value (do at least 100,000 repetitions to find your p-value). 

b. What is the theory-based p-value using the normal distribution? 

c. You should find that your two p-values are not that close. Why do you think they are so different? Which one do you think is more valid? 

d. You have seen Section 1.5 that the normal distribution is an approximation to the way our simulated statistics are distributed. That is why it comes with some validity conditions to make sure the fit is not too bad. The binomial distribution is an exact model for the type of situations we have been exploring throughout Chapter 1 and we can use this distribution to find theory-based p-values for a single proportion. It does not come with the sample size requirements as the normal approximation. Go back to the One Proportion applet and click on the Exact Binomial box. You should now see another p-value based on the binomial distribution. What is that p-value and is it closer to your simulation-based p-value than the normal approximation p-value was?