single trial if the null hypothesis is true. Write down statistical models for $X_1$ and $X_2$ defined in the previous problems under this hypothesis. ##### (b) Under the null hypothesis, what is a statistical model for $X = Xwi + X_2$? Use the combined data from the experiment with all trials with and without a partner to find the maximum likelihood estimate for $p$ assuming the null hypothesis is true. ##### (c) Use simulation to conduct the experiment $3 = 10,G@@$ times using the value for $p$ estimated in the previous problem. This results in $B$ simulated values $XA*_1$ and $XA*_2$ from the assumed statistical model. For each corresponding replicate of the simulation, calculate a test statistic which is the difference in sample proportions. This collection of simulated proportion differences is a simulation-based estimate of the sampling distribution of the test statistic. Find the mean and standard deviation of this distribution. ##### (d) What should the value of the the mean of the sampling distribution approach if we let $B$ approach infinity? What special two-word name is given to the standard deviation of this sampling distribution? ##### (9) Display the distribution of the simulated sampling distribution. Add to this graph a vertical line which is the realized test statistic from the actual data. ##### (f) Calculate the p-value for this hypothesis test. You may either directly report the proportion of extreme simulated proportion differences or make an approximation based on the shape of the sampling distribution to compute an area under an appropriate density curve. ##### (g) Summarize the results of the hypothesis test in context. (In context means you should be discussing what the results say about the probabilities of chimpanzees making prosocial choices with or without partners, and not about statistical significance or rejecting hypotheses.) ### 4 > Write three criticisms of the assumptions made for the previous three hypothesis tests where reality may differ from the assumptions, possibly leading to misleading conclusions