1. Chapter 13 Sampling distributions

2. Objectives (PSLS Chapter 13 & 14)
Sampling distributions
Parameter versus statistic (Awards 27-31)
The law of large numbers (Law of Large Number Award, 28)
Sampling distributions (Sampling Distribution Award, 28)
Sampling distribution of the sample mean (Samp. Distribution Award)
The central limit theorem (Central Limits Theorem Award, 29)

3. Parameter versus statistic
Population: the entire group of individuals in which we are interested but usually can’t assess directly.
A parameter is a number summarizing the population. Parameters are usually unknown.
Sample: the part of the population we actually examine and for which we do have data.
A statistic is a number summarizing a sample. We often use a statistic to estimate an unknown population parameter.
Population
Sample

4. The law of large numbers
Law of large numbers: As the number of randomly drawn observations (n) in a sample increases,
the mean of the sample (x̅) gets closer and closer to the population mean m (quantitative variable).
the sample proportion ( ) gets closer and closer to the population proportion p (categorical variable).
The law of large number has important implications in the gambling and insurance industries.

5. Sampling distributions
Different random samples taken from the same population will give different statistics. But there is a predictable pattern in the long run.
 A statistic computed from a random sample is a random variable.
The sampling distribution of a statistic is the probability distribution of that statistic for samples of a given size n taken from a given population.

6. Note: When sampling randomly from a given population:
The law of large numbers describes what to expect if we took samples of increasing size n.
A sampling distribution describes what would happen if we took all possible random samples of a fixed size n.
Both are conceptual ideas with many important practical applications. We rely on their well known mathematical properties, but we don’t build actual sampling distributions when analyzing data.

7. Sampling distribution of x̅ (the sample mean)
The mean of the sampling distribution of x̅ is μ.
There is no tendency for a sample average to fall systematically above or below μ, even if the population distribution is skewed.
 x̅ is an unbiased estimate of the population mean μ assuming the samples are randomly chosen.
The standard deviation (σ) of the sampling distribution of x̅ is σ/√n.
The standard deviation of the sampling distribution measures how much the sample statistic x̅ varies from sample to sample.
 Averages are less variable than individual observations.

8. For Normally distributed populations
When a variable in a population is Normally distributed, the sampling distribution of the sample mean x̅ is also Normally distributed.
population
Sample means
population N(μ, σ) ↓
sampling distribution N(μ, σ/√n)

9. Deer mice (Peromyscus maniculatus) have a body length (excluding the tail) known to vary Normally, with a mean body length µ = 86 mm, and standard deviation σ = 8 mm.
For random samples of 20 deer mice, the distribution of the sample mean body length is approximately, 
A) Normal, mean 86, standard deviation 8 mm. 
B) Normal, mean 86, standard deviation 20 mm. 
C) Normal, mean 86, standard deviation 1.8 mm. 
D) Normal, mean 86, standard deviation 3.9 mm.

10. Remember that σ is the standard deviation of the original population.
Standardizing a Normal sampling distribution (z)
When the sampling distribution is Normal, we can standardize the value of a sample mean x̅ to obtain a z-score. This z-score can then be used to find areas under the sampling distribution from Table B.
N(µ, σ/√n)
N(0,1)
Here, we work with the sampling distribution, and σ/√n is its standard deviation (indicative of spread).
Remember that σ is the standard deviation of the original population.

11. Hypokalemia is diagnosed when blood potassium levels are low, below 3
Hypokalemia is diagnosed when blood potassium levels are low, below 3.5mEq/dl. Let’s assume that we know a patient whose measured potassium levels vary daily according to iid ~ N(m = 3.8, s = 0.2).
If only one measurement is made, what's the probability that this patient will be diagnosed hypokalemic? Would this be a misdiagnosis?

12. The central limit theorem
Central limit theorem: When randomly sampling from any population with mean m and standard deviation s, when n is large enough, the sampling distribution of x̅ is approximately Normal: N(μ, σ/√n).
The larger the sample size n, the better the approximation of Normality.
This is very useful in inference: Many statistical tests assume Normality for the sampling distribution. The central limit theorem tells us that, if the sample size is large enough, we can safely make this assumption even if the raw data appear non-Normal.

13. How large a sample size?
It depends on the population distribution. More observations are required if the population distribution is far from Normal.
A sample size of 25 or more is generally enough to obtain a Normal sampling distribution from a skewed population, even with mild outliers in the sample.
A sample size of 40 or more will typically be good enough to overcome an extremely skewed population and mild (but not extreme) outliers in the sample.
In many cases, n = 25 isn’t a huge sample. Thus, even for strange population distributions we can assume a Normal sampling distribution of the sample mean, and work with it to solve problems!

14. Population with strongly skewed distribution
Sampling distribution of for n = 2 observations
Sampling distribution of for n = 10 observations
Sampling distribution of for n = 25 observations

15. How do we know if the population is Normal or not?
Sometimes we are told that a variable has an approximately Normal distribution (e.g. large studies on human height or bone density).
Most of the time, we just don’t know. All we have is sample data.
We can summarize the data with a histogram and describe its shape and estimate the likely magnitude of error. We can run simulations to quantify it.
If the sample is random, the shape of the histogram should be similar to the shape of the population distribution.
The central limit theorem can help guess whether the sampling distribution should look roughly Normal or not.

16. (a) Angle of big toe deformations in 38 patients:
Number of subjects
Symmetrical, one small outlier
Population likely close to Normal
Sampling distribution ~ Normal
(b) Histogram of number of fruit per day for 74 adolescent girls
Skewed, no outlier
Population likely skewed
Sampling distribution ~ Normal given large sample size

17. Atlantic acorn sizes (in cm3)
Sample of 28 acorns:
Describe the distribution of the sample. What can you assume about the population distribution?
What would be the shape of the sampling distribution:
For samples of size 1?
For samples of size 5?
For samples of size 15?
For samples of size 50?

18. Objectives (PSLS Chapter 14)
Estimation
Uncertainty and confidence (Margin of Error/CIs Award, 30)
Confidence intervals (Margin of Error/CIs Award, 30)

19. Uncertainty and confidence
If you picked different samples from a population, you would probably get different sample means ( x̅ ) and virtually none of them would actually equal the true population mean, .
Key. Repeated.

20. Use of sampling distributions
Sample means, n subjects
If the population is N(μ,σ), the sampling distribution is N(μ,σ/√n).
If not, the sampling distribution is ~N(μ,σ/√n) if n is large enough.
Population, x individual subjects m
 We can take just one random sample of size n, and rely on the known properties of sampling distributions to estimate the sampling distribution.

21. When we take a random sample, we can compute the sample mean and an interval of size plus-or-minus 2σ/√n about the mean. x̅
Based on the ~ % rule, we can expect that:
Simulation using the applet from
Only the sample circled in green did not capture the population mean μ. All the other intervals in this simulation did capture μ.
~95% of all intervals computed with this method capture the parameter μ.
Red arrow: Interval of size plus or minus 1.96*σ/√n
Blue dot: mean value of a given random sample

22. m falls within the interval with probability (confidence level) C.
Confidence intervals
A confidence interval is a range of values with an associated probability, or confidence level, C. This probability quantifies the chance that the interval contains the unknown population parameter.
m falls within the interval with probability (confidence level) C.

23. The margin of error, m
A confidence interval (“CI”) can be expressed as:
a center ± a margin of error: μ within x̅ ± m
an interval: μ within (x̅ − m) to (x̅ + m)
The confidence level C (in %) represents an area of corresponding size C under the sampling distribution.
Note that not all CI are symmetric about the parameter. Some complex methods produce asymmetric intervals. m m

24. The weight of single eggs varies Normally with standard deviation 5 g
The weight of single eggs varies Normally with standard deviation 5 g. Think of a carton of 12 eggs as an SRS of size 12.
What is the distribution of the sample means ?
You buy one carton of 12 eggs. The average egg weight is x̅ = 64.2g. What can you infer about the mean µ of this population with roughly 95% confidence?

25. CI for a Normal population mean (σ known)
When taking a random sample from a Normal population with known standard deviation σ, a level C confidence interval for µ is:
σ/√n is the standard deviation of the sampling distribution
C is the area under the N(0,1) between −z* and z* C
Use technology or Table C to obtain confidence intervals.
-z* z*
80% confidence level C

26. For 95% confidence level, z* = 1.96 (almost 2)
How do we find z* values?
We can use a table of z and t values (Table C). For a given confidence level C, the appropriate z* value is listed in the same column.
(…)
For 95% confidence level, z* = 1.96 (almost 2)

27. Link between confidence level and margin of error
The confidence level C determines the value of z* (in Table C).
The margin of error also depends on z*.
Higher confidence C implies a larger margin of error m (less precision more accuracy).
A lower confidence level C produces a smaller margin of error m (more precision less accuracy).
 win/loose situation C Z*
−Z*
m m

28. Density of bacteria in solution
Measurement equipment has normal distribution with standard deviation σ = 1 million bacteria/ml of fluid.
3 measurements: 24, 29, and 31 million bacteria/ml.
Mean: = 28 million bacteria/ml. Find the 99% and 90% CI.