(r) prob7a prob7 %>% summarize(mean = mean(p), prob7a sd = sd(p)) Theory and a conceptual understanding of the mean suggest that $\E(p) = 0.5$ as this is the balancing point of this symmetric distribution. Does the simulation support this fact? > [Respond here] Theory and some calculus can be used to show that the standard deviation of $P$ is equal to $1A\sqrt{C}$ where $C$ is an integer. Based on the simulation, what do you think the value of $C$ is? > [Write your answer here] ### 7C Create a graph which shows the observed relative frequency that each for each value of $x$. The sum of these relative frequencies should be one. Compare this plot with the plot of the distribution of $Y \sim \text{Binomial}(9, 0.5)$. How are they similar and how do they differ? Which distribution will have a larger standard deviation based on the graph? (r) > [Write your response here] ### 7D Based on the simulation and the graph of the distribution of $X$, what is your best guess for the exact value of $P(X = 5)$? Compare this probability with the binomial probability $P(Y = 5)$ if $Y \sim \text{Binomial}(9,0.5)$. >[Write your answer here]