Last lecture summary Standard normal distribution, Z-distribution Z-table lognormal distribution, geometric mean

Z-table What is the proportion less than the point with the Z-score -2,75? Nice applet: http://www.mathsisfun.com/data/standard-normal-distribution-table.html

How normal is normal? Checking normality Eyball histograms Eyball QQ plots There are tests http://www.nate-miller.org/blog/how-normal-is-normal-a-q-q-plot-approach

QQ plot Q stands for ‘quantile’. Quantiles are values taken at regular intervals from the data. The 2-quantile is called the median, the 3-quantiles are called terciles, the 4-quantiles are called quartiles (deciles, percentiles).

How to interpret QQ plot

How to interpret QQ plot no outlier no outlier

http://www. nate-miller http://www.nate-miller.org/blog/how-normal-is-normal-a-q-q-plot-approach

Typical normal QQ plot http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

QQ plot of left-skewed distribution http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

QQ plot of right-skewed distribution http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

sampling distributions výběrová rozdělení

Histogram

𝒙 =𝟏𝟗.𝟒𝟒 𝒔=𝟐.𝟒𝟓 𝒏=𝟗 𝒙 =𝟏𝟔.𝟖𝟗 𝒔=𝟗.𝟏𝟕 𝒏=𝟗 𝒙 =𝟏𝟕.𝟐𝟐 𝒔=𝟔.𝟐𝟒 𝒏=𝟗

Sampling distribution of sample mean výběrové rozdělení výběrového průměru

Sweet demonstration of the sampling distribution of the mean

Data 2013 Population: 6,4,5,3,10,3,5,3,6,5,4,8,7,2,8,5,8,5,4,0 20 samples (n=3) and their averages 10 3 5 … 6.0 3 3 4 … 3.3 4 4 8 … 5.3 4 3 8 … 5.0 5 5 6 … 5.3 6 8 7 … 7.0 3 8 8 … 6.3 6 8 4 … 6.0 8 8 4 … 6.7 5 3 4… 4.0 2 10 8… 6.7 3 4 5 … 4.0 5 6 5 … 5.3 8 6 4 … 6.0 4 8 4 … 5.3 5 8 5 … 6.0 4 4 3 … 3.7 8 8 4… 6.7 8 4 5… 5.7 3 0 7… 3.3 http://blue-lover.blog.cz/1106/lentilky

Data 2014 Population: 3,2,3,1,2,6,5,5,4,3,5,5,6,3,2,4,4,3,1,5 20 samples (n=3) and their averages 5 1 4 … 3.3 3 1 1 … 1.7 6 6 5 … 5.7 3 5 4 … 4.0 4 1 4 … 3.0 5 1 3 … 3.0 2 5 4 … 3.7 5 5 1 … 3.7 3 3 5 … 3.7 5 2 3 … 3.3 5 3 4 … 4.0 3 4 6 … 4.3 2 5 5 … 4.0 5 6 1 … 4.0 2 2 5 … 3.0 5 3 6 … 4.7 1 5 3 … 3.0 5 5 5 … 5.0 3 3 6 … 4.0 http://blue-lover.blog.cz/1106/lentilky

Sampling distribution, n = 3 Plot exact sampling distribution sample_size <- 3 data.set2014 <- c(3,2,3,1,2,6,5,5,4,3,5,5,6,3,2,4,4,3,1,5) samps <- combn(data.set2014, sample_size) xbars <- colMeans(samps) barplot(table(xbars))

Sampling distribution, n = 3 Calculate 𝜇. Calculate 𝜎. Le’s create all possible samples of size 3. Calculate 𝑀. Calculate 𝑆𝐸. 𝑆𝐸= 𝜎 𝑛

Sampling distribution, n = 3

Sampling distribution, n = 5

Central limit theorem 𝑀 =𝜇 𝑥 =𝜇 𝑆𝐸= 𝜎 𝑥 = 𝜎 𝑛 Distribution of sample means is normal. The distribution of means will increasingly approximate a normal distribution as the sample size 𝑛 increases. Its mean 𝑀 is equal to the population mean. Its standard deviation 𝑆𝐸 is equal to the population standard deviation divided by the square root of 𝑛. 𝑆𝐸 is called standard error. 𝑀 =𝜇 𝑥 =𝜇 𝑆𝐸= 𝜎 𝑥 = 𝜎 𝑛

Quiz As the sample size increases, the standard error decreases As the sample size increases, the shape of the sampling distribution gets skinnier wider

Another data 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11

Sampling distribution

Sampling distribution

Sampling distribution

Sampling distribution

Sampling distribution applet parent distribution sample data sampling distributions of selected statistics http://onlinestatbook.com/stat_sim/sampling_dist/index.html