Task 1

The chance of 850 or more successes out of 1600 given a 0.5 probability of success.

sum(dbinom(850:1600, 1600, prob=0.5))
## [1] 0.006649478
1/sum(dbinom(850:1600, 1600, prob=0.5))
## [1] 150.3877

The relative likelihood of 800 successes to 850 successes.

dbinom(800, 1600, prob=0.5)/dbinom(850, 1600, prob=0.5)
## [1] 22.76175

Task 2

titanic = read.csv("https://raw.githubusercontent.com/jfcross4/advanced_stats/master/titanic_train.csv")

library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
titanic %>% filter(Age<18) %>% summarize(n=n(), NumSurvived = sum(Survived), SurvivalRate=mean(Survived))
##     n NumSurvived SurvivalRate
## 1 113          61     0.539823
titanic %>% filter(Age<18) %>% group_by(SibSp>=1) %>% summarize(n=n(), NumSurvived = sum(Survived), SurvivalRate=mean(Survived))
## # A tibble: 2 × 4
##   `SibSp >= 1`     n NumSurvived SurvivalRate
##   <lgl>        <int>       <int>        <dbl>
## 1 FALSE           43          28        0.651
## 2 TRUE            70          33        0.471
loners_survived = rbinom(1000, 43, 0.539823)
sibsp_survived = rbinom(1000, 70, 0.539823)

loner_survival_rate = loners_survived/43
sibsp_survival_rate = sibsp_survived/70

difference_in_survival_rate = sibsp_survival_rate - loner_survival_rate

mean(abs(difference_in_survival_rate) > 0.651-0.471)
## [1] 0.06

Task 3

kobe = readRDS(url("https://github.com/jfcross4/advanced_stats/blob/master/kobe_basket.rds?raw=true"))

shots = kobe$shot

shots_after_hits = which(shots[-133]=="H")+1
shots_after_misses = which(shots[-133]=="M")+1
mean(shots=="H")
## [1] 0.4360902
length(shots[shots_after_hits]=="H")
## [1] 57
sum(shots[shots_after_hits]=="H")
## [1] 21
mean(shots[shots_after_hits]=="H")
## [1] 0.3684211
length(shots[shots_after_misses]=="H")
## [1] 75
sum(shots[shots_after_misses]=="H")
## [1] 36
mean(shots[shots_after_misses]=="H")
## [1] 0.48
shots_after_hits = rbinom(1000, 57, 0.4360902)
shots_after_misses = rbinom(1000, 75, 0.4360902)

after_hits_rate = shots_after_hits/57
after_misses_rate = shots_after_misses/75

difference_in_rate = after_hits_rate - after_misses_rate

mean(abs(difference_in_rate) > 0.48 - 0.368)
## [1] 0.194

Task 4

apple = readRDS(url("https://github.com/jfcross4/advanced_stats/blob/master/apple.rds?raw=true"))

mean(apple)
## [1] 0.5352531
days_after_up = which(apple[-5730]==TRUE)+1
days_after_down = which(apple[-5730]==FALSE)+1
length(apple[days_after_up])
## [1] 3067
sum(apple[days_after_up])
## [1] 1598
mean(apple[days_after_up])
## [1] 0.5210303
length(apple[days_after_down])
## [1] 2662
sum(apple[days_after_down])
## [1] 1468
mean(apple[days_after_down])
## [1] 0.5514651
ups_after_ups = rbinom(1000, 3067, 0.5352531)
ups_after_downs = rbinom(1000, 2662, 0.5352531)

ups_after_ups_rate = ups_after_ups/3067
ups_after_downs_rate = ups_after_downs/2662


difference_in_rate = ups_after_ups_rate - ups_after_downs_rate

mean(abs(difference_in_rate) > 0.5514651 - 0.5210303)
## [1] 0.019