• Steve Tipton (3.21) http://rpubs.com/tiptonsteve/DATA606HW3_21
• Betsy Rosalen (3.37) http://rpubs.com/betsyrosalen/Presentation

n <- 1e5
pop <- runif(n, 0, 1)
mean(pop)

## [1] 0.5008915

samp1 <- sample(pop, size=10)
mean(samp1)

## [1] 0.5745289

hist(samp1)

samp2 <- sample(pop, size=30)
mean(samp2)

## [1] 0.5466776

hist(samp2)

M <- 1000
samples <- numeric(length=M)
for(i in seq_len(M)) {
    samples[i] <- mean(sample(pop, size=30))
}
head(samples, n=8)

## [1] 0.5294721 0.4424369 0.5102434 0.4409382 0.5492505 0.5829651 0.5322821
## [8] 0.5063398

Let \(X_1\), \(X_2\), …, \(X_n\) be independent, identically distributed random variables with mean \(\mu\) and variance \(\sigma^2\), both finite. Then for any constant \(z\),

\[ \underset { n\rightarrow \infty }{ lim } P\left( \frac { \bar { X } -\mu }{ \sigma /\sqrt { n } } \le z \right) =\Phi \left( z \right) \]

where \(\Phi\) is the cumulative distribution function (cdf) of the standard normal distribution.

The distribution of the sample mean is well approximated by a normal model:

\[ \bar { x } \sim N\left( mean=\mu ,SE=\frac { \sigma }{ \sqrt { n } } \right) \]

where SE represents the standard error, which is defined as the standard deviation of the sampling distribution. In most cases \(\sigma\) is not known, so use \(s\).

shiny_demo('sampdist')
shiny_demo('CLT_mean')

samp2 <- sample(pop, size=30)
mean(samp2)

## [1] 0.5747245

(samp2.se <- sd(samp2) / sqrt(length(samp2)))

## [1] 0.04941707

The confidence interval is then \(\mu \pm 2 \times SE\)

(samp2.ci <- c(mean(samp2) - 2 * samp2.se, mean(samp2) + 2 * samp2.se))

## [1] 0.4758903 0.6735586

We are 95% confident that the true population mean is between 0.4758903, 0.6735586.

That is, if we were to take 100 random samples, we would expect at least 95% of those samples to have a mean within 0.4758903, 0.6735586.

ci <- data.frame(mean=numeric(), min=numeric(), max=numeric())
for(i in seq_len(100)) {
    samp <- sample(pop, size=30)
    se <- sd(samp) / sqrt(length(samp))
    ci[i,] <- c(mean(samp),
                mean(samp) - 2 * se, 
                mean(samp) + 2 * se)
}
ci$sample <- 1:nrow(ci)
ci$sig <- ci$min < 0.5 & ci$max > 0.5

ggplot(ci, aes(x=min, xend=max, y=sample, yend=sample, color=sig)) + 
    geom_vline(xintercept=0.5) + 
    geom_segment() + xlab('CI') + ylab('') +
    scale_color_manual(values=c('TRUE'='grey', 'FALSE'='red'))

• We start with a null hypothesis (\(H_0\)) that represents the status quo.
• We also have an alternative hypothesis (\(H_A\)) that represents our research question, i.e. what we’re testing for.
• We conduct a hypothesis test under the assumption that the null hypothesis is true, either via simulation or traditional methods based on the central limit theorem.
• If the test results suggest that the data do not provide convincing evidence for the alternative hypothesis, we stick with the null hypothesis. If they do, then we reject the null hypothesis in favor of the alternative.

\(H_0\): The mean of samp2 = 0.5
\(H_A\): The mean of samp2 \(\ne\) 0.5

Using confidence intervals, if the null value is within the confidence interval, then we fail to reject the null hypothesis.

(samp2.ci <- c(mean(samp2) - 2 * sd(samp2) / sqrt(length(samp2)),
               mean(samp2) + 2 * sd(samp2) / sqrt(length(samp2))))

## [1] 0.4758903 0.6735586

Since 0.5 fall within 0.4758903, 0.6735586, we fail to reject the null hypothesis.

\[ \bar { x } \sim N\left( mean=0.49,SE=\frac { 0.27 }{ \sqrt { 30 } = 0.049 } \right) \]

\[ Z=\frac { \bar { x } -null }{ SE } =\frac { 0.49-0.50 }{ 0.049 } = -.204081633 \]

pnorm(-.204) * 2

## [1] 0.8383535