C02_Random_Variable

coin3 <- data.frame( N_Heads = c(0, 1, 2, 3), C_Heads = c(1, 3, 3, 1) ) attach(coin3) coin3$P_Heads <- C_Heads/sum(C_Heads)

options(repr.plot.width=7, repr.plot.height=4) barplot(coin3$P_Heads, names = coin3$N_Heads, main = "Discrete Probability Distribution of flipping three coins")

attach(coin3) E_X = sum(N_Heads * P_Heads) E_X

Uniform_Random_Number <- runif(10000, -2, 1) hist(Uniform_Random_Number, probability = TRUE)

normal_dist <- function(x, b){ mu = mean(x) sigma = sd(x) x1 <- seq(mu - 6*sigma, mu + 6*sigma, length = 100) # Normal curve fun <- dnorm(x1, mean = mu, sd = sigma) # Histogram options(repr.plot.width=7, repr.plot.height=4) hist(x, prob = TRUE, col = "white", breaks = b, xlim = c(mu - 6*sigma, mu + 6*sigma), ylim = c(0, max(fun)), main = "Histogram overlayed with normal curve") lines(x1, fun, col = 2, lwd = 2) }

data(chickwts) attach(chickwts) normal_dist(chickwts$weight, 10)

data(chickwts) attach(chickwts) normal_dist(chickwts$weight, 15)

shapiro.test(chickwts$weight)

chickwts$zscore = (weight - mean(weight))/sd(weight) tail(chickwts)

GRE scores, Part I. Sophia who took the Graduate Record Examination (GRE) scored 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section. The mean score for Verbal Reasoning section for all test takers was 151 with a standard deviation of 7, and the mean score for the Quantitative Reasoning was 153 with a standard deviation of 7.67. Suppose that both distributions are nearly normal.

(a) Write down the short-hand for these two normal distributions.

(b) What is Sophia's Z-score on the Verbal Reasoning section? On the Quantitative Reasoning section?

#b. z_sophiaV = (160 - 151)/7 z_sophiaQ = (157 - 153)/7.67 cat("Z Score for Verbal= ", round(z_sophiaV,2), "\nZ Score for Quantitative Reasoning= ", round(z_sophiaQ, 2))

They tell you how many standard deviations above the mean that she scored on the section of the test

(d) Relative to others, which section did she do better on?

She did better on the Verbal reasoning section since her Z score was better on that section

(e) Find her percentile scores for the two exams.

# e. cat("The verbal ability percentile score is ",round(pnorm(1.29, lower.tail = TRUE),4)*100, "%\n") cat("The quantitative reasoning percentile score is ",round(pnorm(0.52, lower.tail = TRUE),4)*100, "%\n")

(f) What percent of the test takers did better than her on the Verbal Reasoning section? On the Quantitative Reasoning section?

# f. cat(100 - round(pnorm(1.29, lower.tail = TRUE),4)*100, "%", "did better than Sophia in verbal ability\n") cat(100 - round(pnorm(0.52, lower.tail = TRUE),4)*100, "%", "did better than Sophia in quantitative reasoning")

(g) Explain why simply comparing raw scores from the two sections could lead to an incorrect conclusion as to which section a student did better on.

Comparing the raw scores can lead to incorrect conclusions since they are on different scales, while comparing the percentile scores is better to determine how well she did compared to others who took the exam

(h) If the distributions of the scores on these exams are not nearly normal, would your answers to parts (b to (f) change? Explain your reasoning.

It would not change the answer of part b, since the z scores can calculate distributions that are not normal, but the answers for part c-f would change since you can not calculate probabilities with a non-normal model

test <- read.csv("test.csv")