CO2_Random_Variable

coin3 <- data.frame( N_Heads = c(0, 1, 2, 3), C_Heads = c(0, 1, 2, 3) ) 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")

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, 18)

shapiro.test(chickwts$weight)

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

A.Write down the short-hand for these two normal distributions.

a. NVerbal:(μ=151,σ=7) NQuantitativeReasoning:(μ=153,σ=7.67)

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))

C.What do these Z-scores tell you?

D.Relative to others, which section did she do better on?

E.Find her percentile scores for the two exams.

F.What percent of the test takers did better than her on the Verbal Reasoning section? On the Quantitative Reasoning section?

# 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. 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.

This does not let us know how she did compared to other students.

(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.

No, because z-scores are not restricted to only normal distributions.