Total grade: 85/108 (79%)

##########################################################################
# CSC 315, Exam II
# Name: John Palomino 
##########################################################################

##########################################################################
# Add R code to the script below and create a Notebook to explicitly
# answer the following questions. Your Notebook should include output 
# showing the requested results, and written answers to 
# questions should be provided in comments. When you are finished, 
# submit an HTML Notebook through Blackboard as described previously.
#
# Note: DO NOT delete / modify any of the questions / comments below!
##########################################################################

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
library(ggplot2)
library(gtools)

Question 1 -- [8 / 8 points] ✅

# 1) The code below defines a function that randomly simulates drawing 2 cards # from a 52 card deck. The function returns TRUE if the two cards have the # same point value. Use the function to find the empirical probability of # drawing 2 cards with the same value, by calling the function 5000 times. # What is the probability of being dealt two cards with the same point value? # [8 points] draw_2_same <- function() { deck <- rep(1:13,4) s <- sample(deck, 2) s[1] == s[2] } set.seed(123) draw_5000<-replicate(5000,draw_2_same()) draw_5000
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## [4741] FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE
## [4753] FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4765] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4777] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4789] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4801] FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4813]  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4825] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4837] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4849] FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## [4861] FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE  TRUE
## [4873] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4885] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4897] FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## [4909] FALSE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4921] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4933] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4945] FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE
## [4957] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4969] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4981] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4993] FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE
count <- 0
      n2<-5000
      for (i in 1:n2) 
      {
        if(draw_5000[i]== TRUE) 
          count = count+1
      }
      sum.TRUE=count
      sum.TRUE
## [1] 279
prob.TRUE<-sum.TRUE/n2
prob.TRUE
## [1] 0.0558
Question 2 -- [8 / 12 points] ❌

Your if statement is just counting the number of 1s (since 1 is considered TRUE). You want to compare hands[i,1] with hands[i,2].

# 2) The code below generates all possible 2 card hands. Use R to show that the number of # possible 2 card hands is 1,326. [4 points] deck <- rep(1:13,4) hands <- combinations(52, 2, deck, repeats.allowed = FALSE, set = FALSE) length(hands[,1])
## [1] 1326
# 2) Find the theoretical / classical probability of being dealt 2 cards with the 
#    same value, using the 'hands' matrix. [8 points].
draw_2_same <- function() {
  deck <- rep(1:13,4)
  s <- sample(deck, 2)
  s[1] == s[2]
}
count <- 0
      n2<-length(hands[,1])
      for (i in 1:n2) 
      {
        if(hands[i]== TRUE) 
          count = count+1
      }
      sum.TRUE=count
      sum.TRUE
## [1] 126
prob.TRUE<-sum.TRUE/n2
prob.TRUE
## [1] 0.09502262
Question 3 -- [7 / 8 points] ❗

This is the right idea, but is not correct, because you are double counting hands that have 2 aces. To avoid this, you can use sum(hands[,1] == 1 | hands[,2] == 1).

# 3) Find the theoretical / classical probability of being dealt at least one Ace # (which has a point value of 1). [8 points] x=hands[,1] ace1=length(subset(x,x==1)) y=hands[,2] ace2=length(subset(y,y==1)) prob.oneace=(ace1+ace2)/n2 prob.oneace
## [1] 0.1538462
Question 4 -- [0 / 8 points] ❌

You need to apply the calc_score function to each row of hands.

# 4) The function below takes a vector of card values, and finds the total blackjack score, # where face cards (11-13) are worth 10 points and aces (1) are worth 11 points. Use # the function to first find the score of each possible hand, and then to calculate the # probability of being dealt a blackjack (a pair of cards worth 21 points). [8 points] calc_score <- function(myhand) { myhand[myhand > 10] <- 10 # face cards are worth 10 points myhand[myhand == 1] <- 11 # aces are worth 11 points sum(myhand) # find the point total } deck <- rep(1:13,4) deck
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13  1  2  3  4  5  6  7  8  9 10 11 12
## [26] 13  1  2  3  4  5  6  7  8  9 10 11 12 13  1  2  3  4  5  6  7  8  9 10 11
## [51] 12 13
calc_score(deck)
## [1] 380
Question 5 -- [11 / 12 points] ❗

Part (d) is not correct, you need to subtract it from 1, and there is likely a typo. It should be 1 - pnorm(41, 40, 2/sqrt(15)).

# 5) Normal probability calculations. Write the correct R statement to calculate and display # the probability [3 points each = 12 points] # # (a) Suppose that X ~ N(60,5). Find P(X > 64) pnorm(64, mean=60, sd=5, lower.tail = FALSE)
## [1] 0.2118554
#      (b) What is the probability that an observation from the standard normal distribution
#          is greater than 1.24? 
p=1-pnorm(1.24)
p         
## [1] 0.1074877
#      (c) In a population that is normally distributed, calculate the probability 
#          that a randomly selected observation is more than 2 standard deviations
#          BELOW the mean. 
p=pnorm(-2)
p            
## [1] 0.02275013
#      (d) Suppose that X ~ N(40,2) and that 15 individuals are sampled. 
#          Find the probability that the sample mean is at least 41. 
p=pnorm((41-40)/(2/sqrt(15)))
p            
## [1] 0.9735962
Question 6 -- [14 / 16 points] ❗

For (c), n should be 1000, and you should find the p-value from z_statistic; otherwise everything is correct.

# 6) In 2013, the proportion of adults who smoke in the U.S. was 0.18. # A 2019 study involving 1000 adults found that 141 of them smoked. # Is there evidence that the smoking rate has changed? [16 points] # (a) State the null and alternative hypotheses, making sure to define 'p' # H0: p = 0.18 # HA: p!= 0.18, where p = the proportion of adults who smoke in the U.S. # (b) Use the prop.test function to conduct the hypothesis test WITHOUT # the continuity correction. Calculate the z test statistic from the # prop.test object and extract the p-value test=prop.test(141, 1000, p = 0.18,alternative = "two.sided",conf.level = 0.95,correct = FALSE) #computing test statistic from test test$statistic
## X-squared 
##  10.30488
#The z test statistic is 10.305 as shown the results of the test summary.
#extracting p-value
test$p.value         
## [1] 0.00132679
# (c) Calculate the z test statistic using the appropriate formula, and find the p-value 
#     based on this test statistic. Note: these should match your answers from part (b).
n=100
s=141
p=s/n
p0=0.18
z_statistic=(p-p0)/sqrt(p0*(1-p0)/n)  
z_statistic         
## [1] 32.01562
# (d) The correct p-value is approximately 0.00133. Based on this p-value, state the 
#     conclusion regarding the null and alternative hypotheses in the context of this 
#     problem.
#Since the p-value is less than 5 %, we reject the null hypothesis that the smoking rate
#has not changed. Therefore, the smoking rate among US adults has changed between
#2013 and 2019. 
          
Question 7 -- [11 / 12 points] ❗

For (b), you need to take the sqrt of the test statistic. Otherwise, great job!

# 7) According to data from the CDC, for individuals 65 and over, 1968 out of 100,000 have been # hospitalized with COVID-19; for individuals 18-49, 1050 out of 100,000 have been # hospitalized (as of 10/28/2021, source: https://gis.cdc.gov/grasp/COVIDNet/COVID19_3.html). # Is there an association between age and COVID-19 hospitalization based on this data? # [12 points] # (a) State the null and alternative hypotheses, making sure # to define the 'p' parameters # H0: p_65andabove-p_18_49 = 0 # HA: p_65andabove-p_18_49!= 0, where p_65andabove = the proportion of individuals 65 and over hospitalised with COVID-19 # p_18_49 = the proportion of individuals 18-49 hospitalised with COVID-19 # (b) Use the prop.test function to find the test statistic and p-value res <- prop.test(x = c(1968, 1050), n = c(100000,100000)) # Printing the test statistic and p-value res$statistic
## X-squared 
##  282.8934
res$p.value           
## [1] 1.758227e-63
# (c) Your p-value should be very close to 0 (1.758227e-63). Based on this p-value, 
#     state the conclusion regarding the null and alternative hypotheses in the context of 
#     this problem.
# Since the p-value is less than 5%, we reject the null hypothesis that
# there is no association between age and COVID-19 hospitalization. Therefore,
# there is association between age and COVID-19 hospitalization.        
          
Question 8 -- [11 / 12 points] ❗

Great job, you just didn't show the degrees of freedom for (b).

# 8) For this problem we will compare two different insecticides # (sprays) for controlling insects, using the built-in data set # 'Insecticides'. The 'spray' column contains the spray used, and # the 'count' column contains the number of insects in an # experimental area after the spray was used. The code below # filters the data to contain only results from spray 'C' and 'D'. [12 points] sprays <- filter(InsectSprays, spray %in% c("C", "D")) %>% mutate(spray = as.character(spray)) # Our interest is in testing the following hypotheses: # H0: mu_C - mu_D = 0 # HA: mu_C - mu_D != 0, # where mu_C is the mean number of insects in an area sprayed # with spray 'C', and mu_D is the mean number of insects in # an area sprayed with spray 'D'. # (a) Create side-by-side boxplots (using ggplot) showing the # number of insects for each spray. Make sure to label the # axes and give the chart a title. my.bp <<-ggplot(sprays, aes(y= count, x=spray, fill=spray) ) # Creates boxplots my.bp <- my.bp + geom_boxplot() # Adds color my.bp <- my.bp + ggtitle("Distribution of Spray C and D") # Adds a title my.bp <- my.bp + ylab("Count") + xlab("Spray") # Adds kaveks my.bp # displays the boxplots

#   (b) Use the t-test function and display the test statistic, 
#       the degrees of freedom, and p-value. (Note: the sample
#       sample size is small, so we need to assume that the
#       the data is normally distributed for the t-test to be valid, 
#       which we will do here)
test=t.test(count~spray,data=sprays)
test$statistic
##         t 
## -3.078215
test$p.value     
## [1] 0.005729735
#   (c) The p-value should be 0.00573. Given this p-value, state
#        the conclusion regarding the null and alternative hypotheses in 
#        the context of this problem.
#Since p-value is less than 5%, we reject the null hypothesis that there's is no 
#difference in sprays C and D. Hence, sprays C and D differ in effectiveness.       
          
Question 9 -- [10 / 12 points] ❗

For (c) and (d), df = n - 1; otherwise these are correct.

# 9) Consider the following Z test statistics or one sample t-test # statistics for the hypothesis tests discussed in class. Find the # p-value, and state whether you would REJECT the null hypothesis, # or FAIL TO REJECT the null hypothesis based on the p-value. [12 points] # (a) Z = 1.29 2*pnorm(1.29, lower.tail = FALSE)
## [1] 0.1970507
#Fail to reject the null hypothesis because p.value>0.05
# (b) Z = -2.34 (additional question: why would Z be equal to 0)? 
2*pnorm(-2.34)
## [1] 0.01928374
#Reject the null hypothesis because p.value<0.05.
# (c) t = 2.61, n = 45
2*pt(2.61, df = 45, lower.tail = FALSE)
## [1] 0.01225224
#Reject the null hypothesis because p.value<0.05.
# (d) t = -0.29, n = 90
2*pt(-0.29, df = 90)
## [1] 0.7724833
#Fail to reject the null hypothesis because p.value>0.05.

# For questions (9) - (10), state the following:

# (a) What would it mean in the context of this problem if a Type I error occurred?
# For question 9, type I error would occur if we rejected the null hypothesis that 
# there is no difference in sprays C and D if it were true. 
#Question 10
#(a)Reject null hypothesis
#(b)Fail to reject the null hypothesis
#(c)Fail to reject the null hypothesis
#(d)Reject null hypothesis

# (b) What would it mean in the context of this problem if a Type II error occurred?
# For question 9, Type II error would happen if we accept the null 
# hypothesis that there is no difference in sprays C and D. 
#Question 10
#(a)Fail to reject the null hypothesis
#(b)Reject the null hypothesis
#(c)Reject the null hypothesis
#(d)Fail to reject the null hypothesis

Question 10 -- [2 / 4 points] ❌

What do the errors mean in the context of this problem?

# 10) A study is conducted to determine whether the proportion of females in # the United States differs from 50%. The null and alternative hypotheses are # as follows: [4 points] # H0: p = 0.50 # HA: p != 0.50, where p = the proportion of females in the United States # (a) Rejecting H0 if it is true # (b) Accepting H0 if it is false
Question 11 -- [2 / 4 points] ❌

What do the errors mean in the context of this problem?

# 11) A study is conducted to determine whether or not the average age of an adult # male in the U.S. is different than the average age of an adult female. The # null and alternative hypotheses are as follows: [4 points] # H0: mu_male - mu_female = 0 # HA: mu_male - mu_female != 0, where mu_male and mu_female are the mean ages of # adult males and females in the United States. # (a) Rejecting H0 if it is true # (b) Accepting H0 if it is false
Question 12 -- [1 / 0 points] ✅

This is the right start, but you want the quantiles under H0 (when p.hat has mean 0.5, etc).

# 12) Extra Credit - The power of a test is the probability of # correctly rejecting H0 (of rejecting H0 when H0 is false). # The power of a test depends on the sample size (tests # become more powerful as 'n' increases). This should be intuitive. # If we are testing whether a coin is biased, we are much more # likely to have sufficient evidence of bias if we flip the coin # 100 times compared to 10 times. # # Recall that if the true probability of an event is p, then a # sample proportion from a sample of size 'n' has the following # distribution: p.hat ~ N(p, sqrt( p*(1-p)/n)) # # Suppose that the true value of p = 0.60, and we are testing against # H0: p = 0.50, where p = the probability of getting heads. # (a) Use pnorm to calculate the power of the test (the probability of rejecting H0), # when the sample size is 40. Hint: first use qnorm to find the 2.5th and 97.5th quantile # of p.hat, under the null hypothesis. We will call these quantiles 'q1' and 'q2'. # H0 will be rejected if either p.hat < q1 OR p.hat > q2, where p.hat is the # observed sample proportion where p = 0.60. When 'n' is 40, the power is # approximately 0.240. n=40 q1=qnorm(0.025);q1
## [1] -1.959964
q2=qnorm(0.975);q2
## [1] 1.959964
Zleft <- (q1-p)/(sqrt(p*(1-p)/n))
## Warning in sqrt(p * (1 - p)/n): NaNs produced
Zright <-(q2-p)/(sqrt(p*(1-p)/n))
## Warning in sqrt(p * (1 - p)/n): NaNs produced
p<-pnorm(Zright,lower.tail = FALSE)-pnorm(Zleft,lower.tail = TRUE)
p
## [1] NaN
# (b) What type of error would you expect if the true value of p was 0.6 and you
#     conducted a study to reject H0 by flipping a coin 40 times? Why?
# Type I because we mistakenly reject a true null hypothesis

# (c) What type of error would you expect if the true value of p was 0.6 and you
#     conducted a study to reject H0 by flipping a coin 300 times? In this case,
#     the power of the test is 0.938. Why?
# Type I because we mistakenly reject a true null hypothesis