One Minute Paper Results

Homework Presentations

• 3.2 Vic Chan
• 3.3 Ethan
• 3.41 MariAlejandra Ginorio
• 3.43 Michael Ippolito

Coin Tosses Revisited

coins <- sample(c(-1,1), 100, replace=TRUE)
plot(1:length(coins), cumsum(coins), type='l')
abline(h=0)

cumsum(coins)[length(coins)]

## [1] -8

Histogram of Many Random Samples

hist(samples)

Properties of Distribution

(m.sam <- mean(samples))

## [1] 0.236

(s.sam <- sd(samples))

## [1] 10.14096

Properties of Distribution (cont.)

within1sd <- samples[samples >= m.sam - s.sam & samples <= m.sam + s.sam]
length(within1sd) / length(samples)

## [1] 0.678

within2sd <- samples[samples >= m.sam - 2 * s.sam & samples <= m.sam + 2* s.sam]
length(within2sd) / length(samples)

## [1] 0.964

within3sd <- samples[samples >= m.sam - 3 * s.sam & samples <= m.sam + 3 * s.sam]
length(within3sd) / length(samples)

## [1] 0.998

Standard Normal Distribution

$$f\left( x|\mu ,\sigma \right) =\frac { 1 }{ \sigma \sqrt { 2\pi } } { e }^{ -\frac { { \left( x-\mu \right) }^{ 2 } }{ { 2\sigma }^{ 2 } } }$$

x <- seq(-4,4,length=200); y <- dnorm(x,mean=0, sd=1)
plot(x, y, type = "l", lwd = 2, xlim = c(-3.5,3.5), ylab='', xlab='z-score', yaxt='n')

Standard Normal Distribution

Standard Normal Distribution

Standard Normal Distribution

What's the likelihood of ending with less than 15?

pnorm(15, mean=mean(samples), sd=sd(samples))

## [1] 0.9272867

What's the likelihood of ending with more than 15?

1 - pnorm(15, mean=mean(samples), sd=sd(samples))

## [1] 0.07271325

Comparing Scores on Different Scales

SAT scores are distributed nearly normally with mean 1500 and standard deviation 300. ACT scores are distributed nearly normally with mean 21 and standard deviation 5. A college admissions officer wants to determine which of the two applicants scored better on their standardized test with respect to the other test takers: Pam, who earned an 1800 on her SAT, or Jim, who scored a 24 on his ACT?

Standard Normal Parameters

SAT Variability

SAT scores are distributed nearly normally with mean 1500 and standard deviation 300.

• 68% of students score between 1200 and 1800 on the SAT.
• 95% of students score between 900 and 2100 on the SAT.
• 99.7% of students score between 600 and 2400 on the SAT.

Evaluating Normal Approximation

To use the 68-95-99 rule, we must verify the normality assumption. We will want to do this also later when we talk about various (parametric) modeling. Consider a sample of 100 male heights (in inches).

Evaluating Normal Approximation

Histogram looks normal, but we can overlay a standard normal curve to help evaluation.

Normal Q-Q Plot

Skewness

Simulated Normal Q-Q Plots

DATA606::qqnormsim(heights)