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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
1. Eyball histograms
2. Eyball QQ plots
3. 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.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
1.
10 3 5 … 6.0
2.
3 3 4 … 3.3
3.
4 4 8 … 5.3
4.
4 3 8 … 5.0
5.
5 5 6 … 5.3
6.
6 8 7 … 7.0
7.
3 8 8 … 6.3
8.
6 8 4 … 6.0
9.
8 8 4 … 6.7
10. 5 3 4… 4.0
11. 2 10 8… 6.7
12. 3 4 5 … 4.0
13. 5 6 5 … 5.3
14. 8 6 4 … 6.0
15. 4 8 4 … 5.3
16. 5 8 5 … 6.0
17. 4 4 3 … 3.7
18. 8 8 4… 6.7
19. 8 4 5… 5.7
20. 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
1.
5 1 4 … 3.3
2.
3 1 1 … 1.7
3.
6 6 5 … 5.7
4.
3 5 4 … 4.0
5.
4 1 4 … 3.0
6.
5 1 3 … 3.0
7.
2 5 4 … 3.7
8.
5 5 1 … 3.7
9.
3 3 5 … 3.7
10. 5 2 3 … 3.3
11. 5 3 4 … 4.0
12. 3 4 6 … 4.3
13. 2 5 5 … 4.0
14. 5 6 1 … 4.0
15. 2 2 5 … 3.0
16. 5 3 6 … 4.7
17. 1 5 3 … 3.0
18. 5 5 5 … 5.0
19. 3 5 4 … 4.0
20. 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
𝜇.
• Calculate 𝜎 .
• Calculate
• Le’s create all possible samples of size 3.
• Calculate 𝑀.
• Calculate 𝑆𝐸.
𝜎
𝑆𝐸 =
𝑛
Sampling distribution, n = 3
Sampling distribution, n = 5
Central limit theorem
• 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.
𝑀 = 𝜇𝑥 = 𝜇
• Its standard deviation 𝑆𝐸 is equal to the population
standard deviation divided by the square root of 𝑛.
• 𝑆𝐸 is called standard error.
𝜎
𝑆𝐸 = 𝜎𝑥 =
𝑛
Quiz
• As the sample size increases, the standard error
• increases
• 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
n=2
Sampling distribution
n=4
Sampling distribution
n=6
Sampling distribution
n=8
Sampling distribution applet
parent distribution
sample data
sampling distributions
of selected statistics
http://onlinestatbook.com/stat_sim/sampling_dist/index.html