You are to roll a fair die n = 101 times, each time observing if the topside of the die shows a 6 (success) or not (failure). After observing the n = 101 tosses, you are to count the number of times the topside showed a 6. This count is represented by the random variable X.
(a) The distribution of X is Binomial with a mean 16.83 and a standard deviation 3.7454. Enter your answer using all the decimals you can.
(b) Now think about the proportion of your n = 101 tosses that show a six. What can you say about the distribution of this proportion? Complete the sentence, Enter your answer using all the decimals you can.
The distribution of p̂ is approximately Normal with a mean μ_p̂ = 0.1667 and a standard deviation σ_p̂ = 0.03708.
(c) What is the probability that the proportion/percentage of your n = 101 tosses that show a six will be somewhere between 13% and 22%? Enter your answer using all the decimals you can.
0.764
(d) After the n = 101 tosses of the die, you observe X = 23, the value of the sample proportion is then p̂ = 23/101 = 0.2277. What is the probability of observing a sample proportion that is at least this much should you decide to roll this die again 101 times? Enter your answer using all the decimals you can.
0.04363