Now that we have the population mean, we may start taking random samples. I will be showing the concept of the Central Limit Theorem by changing the observations per sample size of 5’s, 10’s, 20’s, and 40’s. We should see a fairly normal approximation distribution as well as a tighter cluster as the sample size increase in each one. I will take 100 samples of each size and we will display them in plots.

fiveSize <- seq(1:100)
tenSize <- seq(1:100)
twentySize <- seq(1:100)
fortySize <- seq(1:100)
str(fiveSize)

 int [1:100] 1 2 3 4 5 6 7 8 9 10 ...

for(i in 1:100)
{
    fiveSize[i] <- sum(rating[sample(1:obs,5)])/5
    tenSize[i] <- sum(rating[sample(1:obs,10)])/10
    twentySize[i] <- sum(rating[sample(1:obs,20)])/20
    fortySize[i] <- sum(rating[sample(1:obs,40)])/40
}

Note The X range*

Note The X range*

Code for all the graphs

fig <- plot_ly(alpha= 0.6)
fig <- fig %>% add_histogram(x = fiveSize, name ="5's" )
fig <- fig %>% add_histogram(x = tenSize, name="10's" )
fig <- fig %>% add_histogram(x = twentySize, name="20's" )
fig <- fig %>% add_histogram(x = fortySize, name="40's")
fig <- fig %>% layout(barmode = "overlay")

fig

fig1 <- plot_ly(alpha= 0.8)
fig2 <- plot_ly(alpha= 0.8)
fig1 <- fig1 %>% add_histogram(x = fiveSize, name="5's")
fig2 <- fig2 %>% add_histogram(x = tenSize, name="10's")

subplot(fig1,fig2)

fig3 <- plot_ly(alpha= 0.8)
fig4 <- plot_ly(alpha= 0.8)
fig3 <- fig3 %>% add_histogram(x = twentySize, name="20s's")
fig4 <- fig4 %>% add_histogram(x = fortySize, name="40's")

subplot(fig3,fig4)

As we can see from the graphs before, the more observations we takke in a given sample, the more clustered the data becomes. The less we take the more spread out the data becomes. Notice that in the 40’s plot we barely go out of the range of the actual mean, with a very small standard deviation.

We will now look at the box plots for the data

Addition of confidence interval, maroon is 5’s, gold is 40’s, and black is the true population mean

allSamples <- c(fiveSize,tenSize,twentySize,fortySize)
categories <- c(replicate(100, "A5"),replicate(100,"B10"),
        replicate(100,"C20"),replicate(100,"D40"))
dataRatings <- data.frame(allSamples, categories)

p <- ggplot(dataRatings, aes(x = categories, y = allSamples,
                 fill = categories)) +
geom_boxplot(notch = T)

p

sampMeanFi  <- mean(fiveSize)
sampMeanFo  <- mean(fortySize)
sampSDFi <- sd(fiveSize)
sampSDFo <- sd(fortySize)

upConfFi <- sampMeanFi + 1.96*sampSDFi/sqrt(length(fiveSize))
lowConfFi <- sampMeanFi - 1.96*sampSDFi/sqrt(length(fiveSize))

upConfFo <- sampMeanFo + 1.96*sampSDFo/sqrt(length(tenSize))
lowConfFo <- sampMeanFo - 1.96*sampSDFo/sqrt(length(tenSize))

p <- p + geom_hline(yintercept=popMean, color="black") +
     geom_hline(yintercept=upConfFi, color="maroon",linetype="dashed") +
     geom_hline(yintercept=lowConfFi, color="maroon",linetype="dashed") +
     geom_hline(yintercept=upConfFo, color="gold",linetype="dashed") +
     geom_hline(yintercept=lowConfFo, color="gold",linetype="dashed")

p

• Confidence interval

\[ \bar{x} \pm Z_{\alpha \over 2} \cdot {sd \over \sqrt{n} }\] - Sample Standard Deviation

\[\sqrt{\sum_{i=1}^n (\bar{x} - x_i)^2 \over n-1}\]

As we can see, increasing obesrvation count is crucial to having more accurate results. Both the Law of Large Numbers and Central Limit Theorem play a huge roll in society and this trivial example shows how powerful they are.

Credits go to: