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

2. Z-table
What is the proportion less than the point with the Z-score -2,75?
Nice applet:

3. How normal is normal? Checking normality Eyball histograms
Eyball QQ plots
There are tests

4. 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).

5. How to interpret QQ plot

6. How to interpret QQ plot
no outlier
no outlier

7. http://www. nate-miller

8. Typical normal QQ plot

9. QQ plot of left-skewed distribution

10. QQ plot of right-skewed distribution

11. sampling distributions
výběrová rozdělení

13. Histogram

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

15. Sampling distribution of sample mean
výběrové rozdělení výběrového průměru

16. Sweet demonstration of the sampling distribution of the mean

17. 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
… 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

18. 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

19. 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))

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

21. Sampling distribution, n = 3

22. Sampling distribution, n = 5

23. 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

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

25. 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

26. Sampling distribution

27. Sampling distribution

28. Sampling distribution

29. Sampling distribution

30. Sampling distribution applet
parent distribution
sample data
sampling distributions of selected statistics