Problem 1

studio7_problem_1(0.7, 0.05, 18)
## ----------------------------------
## Problem 1: Rejection region, actual significance, power
##   Rejection region:  13 14 15 16 17 18 
##   True significance= 0.04812622 
##   Power= 0.5343801

Problem 2

studio7_problem_2(0.7, 0.05, 18, 1000, c(0.3,0.7))
## ----------------------------------
## Problem 2: Simulation with a mixture of coins
##   theta_HA=0.7, alpha=0.05
##   n_tosses=18, n_trials=1000
##   secret_prior= 0.3 0.7 
##   Number of rejections:  391 
##   Number of type 1:  5 
##   Number of type 2:  306 
##   P(rejection | H0):  0.01623377 
##   P(H0 | rejection):  0.01278772 
##   P(rejection | HA):  0.5578035 
##   P(HA | rejection):  0.9872123 
##   P(rejection):  0.391

Problem 3

Problem 3a

studio7_problem_3a(0.7, 0.05, 18, 1000)
## -----
## 3a: Simulation with all fair coins: explain results
## ----
## ----------------------------------
## Problem 2: Simulation with a mixture of coins
##   theta_HA=0.7, alpha=0.05
##   n_tosses=18, n_trials=1000
##   secret_prior= 1 0 
## 
##  Giving the actual output from this function will give too much away. So we blank out the answers with ???
##   Number of rejections:  ??? 
##   Number of type 1:  ??? 
##   Number of type 2:  ??? 
##   P(rejection | H0):  ??? 
##   P(H0 | rejection):  ??? 
##   P(rejection | HA):  ??? 
##   P(HA | rejection):  ??? 
## # Problem 2 called from problem 3a----
##   You need to edit the cat statements at the end of studio7_problem_3a() to give a short explanation of the estimated probabilities that go with the given prior.

Problem 3b

studio7_problem_3b(0.7, 0.05, 18, 1000)
## -----
## 3b: Simulation with all ubfair coins: explain results
## ----
## ----------------------------------
## Problem 2: Simulation with a mixture of coins
##   theta_HA=0.7, alpha=0.05
##   n_tosses=18, n_trials=1000
##   secret_prior= 0 1 
## 
##  Giving the actual output from this function will give too much away. So we blank out the answers with ???
##   Number of rejections:  ??? 
##   Number of type 1:  ??? 
##   Number of type 2:  ??? 
##   P(rejection | H0):  ??? 
##   P(H0 | rejection):  ??? 
##   P(rejection | HA):  ??? 
##   P(HA | rejection):  ??? 
## # Problem 2 called from problem 3b----
##   You need to edit the cat statements at the end of studio7_problem_3b() to give a short explanation of the estimated probabilities that go with the given prior.

Problem 3c

studio7_problem_3c()
## -----
## 3c: Explain difference between P(H0 | rejection) and significance
##   You need to edit the cat statements at the end of studio7_problem_3c() to explain the difference asked about.

Problem 3d

studio7_problem_3d()
## -----
## 3d: Shout it out!
##   You need to edit the cat statements at the end of studio7_problem_3d() to shout and say what is asked.

Problem 4 is OPTIONAL

studio7_problem_4(0.7, 0.05, 18, c(0.1, 0.9))
## ----------------------------------
## OPTIONAL 4: Use Bayes theorem
##   true P(H0 | rejection):  0.009907515 
##   true P(HA | rejection):  0.9900925
# Compare the problem 4 results with the simulated results in problem 2.
studio7_problem_2(0.7, 0.05, 18, 10000, c(0.1, 0.9))
## ----------------------------------
## Problem 2: Simulation with a mixture of coins
##   theta_HA=0.7, alpha=0.05
##   n_tosses=18, n_trials=10000
##   secret_prior= 0.1 0.9 
##   Number of rejections:  4918 
##   Number of type 1:  54 
##   Number of type 2:  4181 
##   P(rejection | H0):  0.0565445 
##   P(H0 | rejection):  0.01098007 
##   P(rejection | HA):  0.5377557 
##   P(HA | rejection):  0.9890199 
##   P(rejection):  0.4918

MIT OpenCourseWare

https://ocw.mit.edu

18.05 Introduction to Probability and Statistics

Spring 2022

Authors: Jeremy Orloff and Jennifer French

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