Download sampling distribution

SUMMARY
1.
2.
3.
My rating is 1800.
8110th place among world competitive chess players.
Ranked higher than 88% of competitive chess players.
We should use relative
frequencies and convert
all absolute frequencies
to proportions.
Distribution of scores in one particular year
𝑥 = 173 cm
𝑠 = 5 cm
-2
-1
0
+1
+2
Z-score
• Standardize normal distribution by converting any point
using this formula
𝑥−𝜇
𝑍=
𝜎
• Standard normal distribution … N(0,1)
NEW STUFF
Quiz
• Approximately what proportion of people is smaller than
168 cm?
𝑥 = 173 cm
𝑠 = 5 cm
16%
163
168
173
178
183
Quiz
• Approximately what proportion of people is higher than
183 cm?
𝑥 = 173 cm
𝑠 = 5 cm
2.5%
163
168
173
178
183
Quiz
• Approximately what proportion of people is between 163
cm and 178 cm high?
𝑥 = 173 cm
𝑠 = 5 cm
81.5%
163
168
173
178
183
Quiz
• Approximately what proportion of people is smaller than
180 cm?
𝑥 = 173 cm
𝑠 = 5 cm
ca 91.5%
163
168
173
178
183
Quiz
• What is the probability of randomly selecting a height in
the sample that is >5 standard deviations above the
mean?
1.
2.
3.
4.
0.01
0.3
0.8
0.99
Quiz
• What is the probability of randomly selecting a height in
the sample that is <5 standard deviations below the
mean?
1.
2.
3.
4.
0.01
0.3
0.8
0.99
Quiz
• What proportion of the data is either below 2 standard
deviations or above 2 standard deviations from the mean
for a normal distribution?
95%
2.5%
2.5%
Z-table
What is the proportion less than the point with the Z-score -2,75?
Use Z-table
What proportion of people is smaller than 180 cm?
𝑥 = 173 cm
𝑠 = 5 cm
180 − 173 7
Z − value =
= = 1.4
5
5
Z-value of 1.4 corresponds to 91.92%.
Quiz – height data
• 𝑛 = 1000, 𝑥 = 173, 𝑠 = 5.0
• What proportion of people is smaller than you?
• 𝑍 =
??−173
5
= , proportion = see Z-table
• What proportion of people is taller than you?
• 𝑍=
??−173
5
= , proportion = 1 − see Z-table
• Table gives a value “less than”.
• Note, that “greater than x” is the same as “less than -x”.
Quiz – height data
• 𝑛 = 1000, 𝑥 = 173, 𝑠 = 5.0
• What proportion of people lie between you and you?
• 𝑍𝐴 =
𝑎−173
5
=, 𝑍𝑏 =
proportion(𝑍𝑏 ) =
𝑏−173
5
=, proportion = proportion 𝑍𝑎 −
• How tall should you be to be in the top 5% of the highest
people?
• 𝑍 − 𝑠𝑐𝑜𝑟𝑒 = 1.645, 173 + 1.645 × 5.0 ≈ 181 cm
An intriguing fact
𝑥 = 173 cm
𝑠 = 5 cm
SAMPLING
DISTRIBUTIONS
Tetrahedral die
http://www.mtgfanatic.com/store/dice/dice.aspx?catid=355
Some statistics
• What is the population?
• 1, 2, 3, 4
• What is the population mean?
• 𝜇=
1+2+3+4
4
=
10
4
= 2.5
• 𝜇 is also called the expected value. We don’t really expect to get
2.5 – it’s not even the option. We expect to get somewhere around
2.5 if we take the sample from this population.
• If we roll the dice twice, how many possible outcomes we
can get?
• 16
16 samples, 𝑛 = 2
1,1
1,2
1,3
1,4
𝒙=𝟏
𝒙 = 𝟏. 𝟓
𝒙=𝟐
𝒙 = 𝟐. 𝟓
2,1
2,2
2,3
2,4
𝒙 = 𝟏. 𝟓
𝒙=𝟐
𝒙 = 𝟐. 𝟓
𝒙=𝟑
3,1
3,2
3,3
3,4
𝒙=𝟐
𝒙 = 𝟐. 𝟓
𝒙=𝟑
𝒙 = 𝟑. 𝟓
4,1
4,2
4,3
4,4
𝒙 = 𝟐. 𝟓
𝒙=𝟑
𝒙 = 𝟑. 𝟓
𝒙=𝟒
What is the mean of the sample means? We design it 𝑀.
Some guess?
Copy sample means from my website (section data) to the WolframAlpha
(link is also provided at my website).
𝑀 = 2.5 = 𝜇
Sampling distribution
Distribution of sample means – sampling distribution
(výběrové rozdělení, výběrové rozdělení výběrového
průměru)
What’s the shape?
•
•
•
•
Uniform
Bimodal
Normal
Skewed
http://www.wolframalpha.com/input/?i=1%2C+1.5%2C+2%2C+2.5%2C+1.5%2C+2%2C+2.5%2C+3%2C+2%2C+2.5%2C+3%2C+3.5%2C+2.5%2C+3%2C+3.5%2C+4
Quiz
What’s the probability that the average of your two rolls will
be 3 or more?
3+2+1
= 0.375
16
http://www.wolframalpha.com/input/?i=1%2C+1.5%2C+2%2C+2.5%2C+1.5%2C+2%2C+2.5%2C+3%2C+2%2C+2.5%2C+3%2C+3.5%2C+2.5%2C+3%2C+3.5%2C+4
Real life
• We can easily calculate the probability for discrete
samples in a discrete population.
• What about in real life? We have huuuuge populations
(e.g. 3 000 000). We can’t calculate means of all possible
samples.
1
1.5
2
2.5
3
3.5
4
Standard deviation of the sample means
• Calculate the population standard deviation
𝜎 for our
population (1, 2, 3, 4).
• Answer: 𝜎 = 1.118, do not forget to divide by 𝑛!
• Now calculate the standard deviation of the sampling
distribution 𝑆𝐸.
1,1 𝒙 = 𝟏
2,1 𝒙 = 𝟏. 𝟓
3,1 𝒙 = 𝟐
4,1 𝒙 = 𝟐. 𝟓
1,2 𝒙 = 𝟏. 𝟓
2,2 𝒙 = 𝟐
3,2 𝒙 = 𝟐. 𝟓
4,2 𝒙 = 𝟑
1,3 𝒙 = 𝟐
2,3 𝒙 = 𝟐. 𝟓
3,3 𝒙 = 𝟑
4,3 𝒙 = 𝟑. 𝟓
1,4 𝒙 = 𝟐. 𝟓
2,4 𝒙 = 𝟑
3,4 𝒙 = 𝟑. 𝟓
4,4 𝒙 = 𝟒
• I did it for you. Before showing you the answer, did I divide by 𝑛 or
by 𝑛 − 1?
• By 𝑛. And 𝑆𝐸 = 0.790.
Relationship between 𝜎 and 𝑆𝐸
• Is there any relationship between
1. 𝜎 = 1.118 and
2. 𝑆𝐸 = 0.790 ?
• Be prepared for a big surprise!
• What is the ratio
𝜎
?
𝑆𝐸
Someone calculate this number for
me, please.
Population standard
deviation
𝜎
= 𝑛
𝑆𝐸
Standard deviation of distribution of sample
means (sampling distribution)
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 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.
• Distribution we draw samples from can be of any shape.
And still sampling distribution of the mean is normal.
Quiz
• As the sample size increases, the standard error
𝜎
𝜎
• increases
= 𝑛 ⟹ 𝑆𝐸 =
𝑆𝐸
𝑛
• decreases
• As the sample size increases, the shape of the sampling
distribution gets
• skinnier
• wider