3.17 Scores on stats final.

  1. The mean score is 77.7 points. with a standard deviation of 8.44 points. Use this information to determine if the scores approximately follow the 68-95-99.7% Rule.
score <- c(57,66,69,71,72,73,74,77,78,78,79,79,81,81,82,83,83,88,89,94)
score_mean <- 77.7
score_sd <- 8.44

#1sd
sd1Upp = score_mean + score_sd
sd1Upp
## [1] 86.14
sd1Low = score_mean - score_sd
sd1Low
## [1] 69.26
#2sd
sd2Upp = score_mean + (2*score_sd)
sd2Upp
## [1] 94.58
sd2Low = score_mean - (2*score_sd)
sd2Low
## [1] 60.82
#3sd
sd3Upp = score_mean + (3*score_sd)
sd3Upp
## [1] 103.02
sd3Low = score_mean - (3*score_sd)
sd3Low
## [1] 52.38
##68 rule
length(score[score >= sd1Low & score <= sd1Upp] ) / length(score) * 100
## [1] 70
##95 rule
length(score[score >= sd2Low & score <= sd2Upp] ) / length(score) * 100
## [1] 95
##99.7 rule
length(score[score >= sd3Low & score <= sd3Upp] ) / length(score) * 100
## [1] 100

Ans: It shows that the scores approximately follow the 68-95-99.7 rule. 70% of data sets are within 1 standard deviation, 95% fall within 2 standard deviations and 100% fall within 3 standard deviations.



  1. Do these data appear to follow a normal distribution? Explain your reasoning using the graphs provided below.
hist(score, probability = TRUE, col = "sky blue")

x <- 50:100
y <- dnorm(x=x, mean = score_mean, sd = score_sd)
lines(x=x, y=y, col = "red")

qqnormSim(score)

Ans: The distribution looks pretty like much bell-shaped, unimodal and symmetric as 68-95-99.7 rule works in this case. Also, Q-Q plot indicates that most of data points are approximately fitting the straight line. Therefore, we can say that data follows a normal distribution.