Problem 1

Problem 1a

rquiz_practice_problem_1a(8, 0, 1)
## ----------------------------------
## Problem 1a: sampling from a normal distribution
## 2.228374 -0.3991196 1.02897 0.2104928 -1.324092 -1.502721 1.143419 -0.2782333

Problem 1b

rquiz_practice_problem_1b(10, 10, 0.3)
## -----
## 1b: sampling from a binomial distribution
## 5 3 1 3 3 4 2 3 5 3

Problem 1c

sample_space = c('Yes', 'No', 'Maybe')
rquiz_practice_problem_1c(sample_space, 7)
## -----
## 1c: sampling from a list with replacement
## Maybe Maybe No Yes Yes Yes Yes

Problem 1d

rquiz_practice_problem_1d(6)
## -----
## 1d: Permutations
## 3 4 1 5 2 6

Problem 1e

rquiz_practice_problem_1e(2, 2*pi)
## -----
## 1e: Plotting

## See plot: Plot for problem 1e

Problem 1f

rquiz_practice_problem_1f(5000, 1, 5)
## -----
## 1f: Sampling and basic statistics
## mean: 3.010489 
## median: 3.002567 
## var: 1.293794 
## sd: 1.137451 
## 0.25 and 0.75 quantiles for the data: 2.045466 3.991788

Problem 1g

rquiz_practice_problem_1g(50, 1,3)
## -----
## 1g: Sampling and basic statistics
## Covariance of x an z: 0.1393713 
## Correlation of x an z: 0.6616549

## See plot: Scatterplot for problem 1g

Problem 1h

rquiz_practice_problem_1h(50, 0, 2, 1.5, 2, 0.05, 0, 2)
## -----
## 1h: Sampling and basic statistics
## 1h: n_samples=50, alpha=0.05, mu0=0, sigma0=2
## mu1=0, sigma1=2, mu2=1.5, sigma2=2
## ---Problem 1h(i) 
## z_stat = -0.8976517 
## p_value = 0.3693713 
## The data does not support rejecting H0:mu1 = 0(assuming sigma1=2)
## ---Problem 1h(ii) 
## t_stat = -0.8987041 
## p_value = 0.3732054 
## The data does not support rejecting H0: mu1 = 0 
## ---Problem 1h(iii) 
## t_stat = -5.050312 
## p_value = 2.040877e-06 
## Reject H0 in favor of HA: mu1 != mu2
## ---Problem 1h(iv) 
## test_stat = 0.8535616 
## p_value = 0.5815822 
## The data does not support rejecting H0: var(x1) == var(x2)
rquiz_practice_problem_1h(50, 0, 2, 0, 2, 0.05, 0, 2)
## -----
## 1h: Sampling and basic statistics
## 1h: n_samples=50, alpha=0.05, mu0=0, sigma0=2
## mu1=0, sigma1=2, mu2=0, sigma2=2
## ---Problem 1h(i) 
## z_stat = -1.752476 
## p_value = 0.07969205 
## The data does not support rejecting H0:mu1 = 0(assuming sigma1=2)
## ---Problem 1h(ii) 
## t_stat = -2.134783 
## p_value = 0.03780555 
## Reject H0 in favor of HA: The mean is not equal to 0 (assuming sigma1= 2 )
## ---Problem 1h(iii) 
## t_stat = -1.944973 
## p_value = 0.05464438 
## The data does not support rejecting H0 that: mu1 = mu2
## ---Problem 1h(iv) 
## test_stat = 0.5583306 
## p_value = 0.04386132 
## Reject H0 in favor of HA: var(x1) != var(x2)
rquiz_practice_problem_1h(50, 1, 5, 1.5, 2, 0.05, 0, 2)
## -----
## 1h: Sampling and basic statistics
## 1h: n_samples=50, alpha=0.05, mu0=0, sigma0=2
## mu1=1, sigma1=5, mu2=1.5, sigma2=2
## ---Problem 1h(i) 
## z_stat = 3.477764 
## p_value = 0.0005056143 
## Reject H0 in favor of HA: The mean is not equal to 0 (assuming sigma1= 2 )
## ---Problem 1h(ii) 
## t_stat = 1.261255 
## p_value = 0.2131889 
## The data does not support rejecting H0: mu1 = 0 
## ---Problem 1h(iii) 
## t_stat = -0.03302583 
## p_value = 0.9737212 
## The data does not support rejecting H0 that: mu1 = mu2
## ---Problem 1h(iv) 
## test_stat = 10.33645 
## p_value = 1.170175e-13 
## Reject H0 in favor of HA: var(x1) != var(x2)

Problem 2

rquiz_practice_problem_2(c(.4,.2,.2,.1,.1), 4)
## ----------------------------------
## Problem 2: Simulation with a mixture of dice
## prior = 0.4 0.2 0.2 0.1 0.1 
## chosen die = 6 
## roll = 5 
## posterior = 0 0.4651163 0.3488372 0.1162791 0.06976744 
## Posterior prediction: P( 4 | data) = 0.1343023
rquiz_practice_problem_2(c(.4,.2,.2,.1,.1), 12)
## ----------------------------------
## Problem 2: Simulation with a mixture of dice
## prior = 0.4 0.2 0.2 0.1 0.1 
## chosen die = 4 
## roll = 3 
## posterior = 0.5825243 0.1941748 0.1456311 0.04854369 0.02912621 
## Posterior prediction: P( 12 | data) = 0.005501618

Problem 3

Problem 3a

rquiz_practice_problem_3a(3)
## ----------------------------------
## Problem 3a: counting Brass rats
## 3 rings can be worn 6 different ways
rquiz_practice_problem_3a(4)
## ----------------------------------
## Problem 3a: counting Brass rats
## 4 rings can be worn 12 different ways

Problem 3b

rquiz_practice_problem_3b(30000)
## ----------------------------------
## Problem 3b: more counting Brass Rats
## They would need at least 174 Brass rats.

Problem 4

Problem 4a

rquiz_practice_problem_4a(200, 0.7)
## ----------------------------------
## Problem 4a: Plot binomial

## See plot: Plot for 4a

Problem 4b

rquiz_practice_problem_4b(200, 129, 0.7, 0.05)
## -----
## Problem 4b: NHST
## H0: Support for Tim is 0.7 
## HA: support for Tim is less than 0.7 
## rejection region: (left tail): x <= 128 
## p = 0.05420741 
## Do not reject H0
rquiz_practice_problem_4b(200, 128, 0.7, 0.05)
## -----
## Problem 4b: NHST
## H0: Support for Tim is 0.7 
## HA: support for Tim is less than 0.7 
## rejection region: (left tail): x <= 128 
## p = 0.03962828 
## Reject H0

Problem 5

Problem 5a

rquiz_practice_problem_5a(0.2)
## ----------------------------------
## Problem 5a: Plot exponential

## See plot: Plot for problem 5a

Problem 5b

rquiz_practice_problem_5b(0.2, 1000)
## -----
## Problem 5b: Simulation

## See plot: Problem 5b(i): frequency histogram of data

## See plot: Problem 5b(ii): density histogram of data

Problem 6

Problem 6a

rquiz_practice_problem_6a(40, 7)
## ----------------------------------
## Problem 6a: Theoretical mean and variance
## one_bet_mu = -0.05263158 
## one_bet_var = 0.9972299 
## Expected winnings in one day = -2.105263 
## Expected variance of one day's winning = 39.8892 
## Expected winnings in 7 days = -14.73684 
## Expected variance of 7 winnings in = 279.2244

Problem 6b

rquiz_practice_problem_6b(40, 8, 10000)
## -----
## Problem 6b: Simulation

## See plot: Problem 6b(i): histogram of 8 days winnings 
## Explanation: Each experimental trial gives the sum of n_days_bets independent bets.
## One bet has mean one_bet_mu and variance one_bet_var.
## So, the Central Limit Theorem, one trial is approximately normal with mean
##      n_bets_per_trial*one_bet_mu
## and variance
##      n_bets_per_trial*one_bet_var

Problem 7

set.seed(1)
mit_times = rnorm(25, 15, 4)
harvard_times = rnorm(25, 18, 5)
rquiz_practice_problem_7(mit_times, harvard_times, 0.05)
## ----------------------------------
## Problem 7: Using google
## H0: the median speeds of the two schools is the same.
## HA: the median speeds of the two schools are different.
## The median times for MIT and Harvard were 16.55937 and 17.70343 respectively.
## test stat = 208 
## p_value = 0.04289115 
## Reject H0: the median speeds of the two schools differed at significance level 0.05 
## Since the median MIT time is less than that of Harvard, it appears that MIT is the faster school.

MIT OpenCourseWare

https://ocw.mit.edu

18.05 Introduction to Probability and Statistics

Spring 2022

Authors: Jeremy Orloff and Jennifer French

For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms