1. Probability & Statistical Inference Lecture 4
MSc in Computing (Data Analytics)

2. Lecture Outline Recap to Statistical Inference Central Limit Theorem
Confidence Intervals
Section Takeaways

3. Statistical Analysis Process
Population
Representative Sample
Sample Statistic
Make Inference
Describe

4. Populations vs. Samples
How do Irish voters intend voting in the next election?
The voting population of Ireland:
2,680,0001
A sample of 1,008 adults was taken and surveyed for their voting intention in the next election2
Source -

5. Populations vs. Samples
How do Irish voters intend voting in the next election?
1,008 voters were asked how they intended to vote in the next election
Fine Gael: 30%
Labour: 14%
Fianna Fail: 18%
Sinn Fein: 17%
Other: 21%

6. Populations vs. Samples
The term population is used in statistics to represent all possible measurements or outcomes that are of interest to us in a particular study or piece of analysis
In the example the population of interest was the voting intentions of all voters in Ireland
The term sample refers to a subset of the population that is selected for analysis
In the example the polling company selected a sample of 1,008 voters

7. Sampling
In choosing a sample it is important that it is representative of the population
No bias should exist in the sample
There are a number of sampling methods available to ensure that your data is representative
A simple random sample is the most straight forward of these methods

8. Statistical Inference
The statistical methods used to draw conclusions about populations based on the statistics describing a sample is known as statistical inference
We want to make decisions based on evidence from a sample i.e. extrapolate from sample evidence to a general population
To make such decisions we need to be able to quantify our (un)certainty about how good or bad our sample information is

9. Statistical Inference
Statistical Inference is divided into two major areas:
Parameter Estimation: This is where sample statistics are used to estimate population parameters
Hypothesis Testing: A statistical hypothesis is a statement about the parameters of one or more populations. Hypothesis testing tests whether a hypothesis is supported by data collected

10. Population Statistics – Point Estimation
The population mean is denoted by µ (mu)
In general, given a sufficiently large sample, we use the sample mean as a point estimate of µ
The population variance is denoted by σ2 (sigma- squared)
In general, given a sufficiently large sample, we use the sample variance s2 as a point estimate of σ2

11. Population Statistics – Point Estimation
An estimate of proportion, p, of items in a population that belong to a class of interest is calculated as:
where c is the number of items in a random sample of size n that belong to the class of interest
This is known as the sample proportion

12. Central Limit Theorem

13. Demonstration

14. Central Limit Theorem Explained by Example
The distribution shown is a poission distribution with λ=3
This could represent the distribution of the number of clicks on a particular link in one second

15. Create 200 sample distributions each with a large sample size
Calculate the mean of each distribution

16. Central Limit Theorem Explain what has happened?
As the sample sizes increased the shape of the histogram of means tended towards a normal distribution
As the sample sizes increased the spread (standard deviation) between the sample means decreased

17. Central Limit Theorem
These histograms are pictures of The Sampling Distribution of the Mean
This phenomenon will happen in ALL cases
The proof of this is called the Central Limit Theorem (CLT) and involves some fairly non-trivial mathematics

18. Definition: Central Limit Theorem continued…
The sampling distribution of the mean has a average value =  (the population mean).
The sampling distribution of the mean has a standard deviation
Where σ is the population standard deviation, and n is the sample size taken.
This value is called the standard error of the mean.
The Sampling Distribution of the Mean will be a Normal distribution if the sample size is large.

19. Central Limit Theorem - Definition
If a random sample is taken from a population, where:
Each member of the sample can be considered to be independent of each other
The are all members of the same population
That population has a mean value μ and a standard deviation σ
Then

20. Central Limit Theorem - Definition
This is a non-mathematical definition of the Central Limit Theorem (CLT)
The central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean μ and a variance σ²/n as n, the sample size, increases

21. The Distribution of the Sample Means

22. Confidence Intervals

23. How can we use the CLT
The Central Limit Theorem avoids the necessity of specifying a complete statistical model for all the sampled data.
All we have to do is specify a probability model for the sample mean.
For any sample mean, calculated from a large independent random sample taken from any population with a mean μ and standard deviation σ, we know from the CLT, that this sample mean is a random variable from a Normal distribution with a mean = μ and a standard deviation =

24. Practical use for the CLT continued…
Take a single sample and calculate
This is an estimate of μ – the true (but unknown) population mean.
But, how good is this estimate?
We assume that is not exactly , but  is somewhere near - but how near is it likely to be?

25. Confidence Intervals Introduction
We would like to make probability statements as to how close is likely to be to .
If sample size is sufficiently large – then the estimate
can be considered as:
a random variable from a Normal distribution,
so probability statements are possible.
This is how we use the CLT in practical data analysis.

26. Confidence Intervals Introduction
For a Normal distribution, we know that 95% of values will be within 1.96 Standard deviations of 
So, given one estimate we can say that this estimate is within 1.96 standard errors of the actual population mean , with 95% confidence
95% in shaded area
We can turn this knowledge on its head: given we can be 95% confident that the true mean  is within 1.96 standard errors of it.

27. Confidence Interval
From this we can specify a range of values within which we are 95% confident that the population mean () lies
This is called a confidence interval
95% Confidence Interval for a population mean
(from large enough sample):
Remarkably, this result holds for samples of size 30 or more. So, a large sample in this context, is a sample of 30 or more.

28. Example
One sample of size 30 from the electronic components yields a sample mean = 5,873 hours .We know  = 3,959 so a 95% confidence interval would be:
Interpretation: we would say that the average lifetime of all components (μ) is between 4,456 and 7,290 hours with 95% confidence

29. Confidence Intervals Why is this any good?
Before: one estimate, = 5,873 but no idea of how good or bad it was, i.e. how close to μ is was likely to be.
Now: 95% confident that μ is between 4,456 and 7,290 hours.
So, using CLT leads to Confidence Intervals that enables us to estimate a statistic with certain level of confidence.
In other word it gives us an objective measure of the actual amount of information contained in our sample about the likely location of μ.

30. Problem with σ
All of the above assumes that the population standard deviation (i.e. ) is known.
In practice this is not known (just like ).
So, we need to estimate  as well as 
we get this estimate from the standard deviation of the sample, given that the sample is large enough.
Sample Standard Deviation is called ‘s’
Estimate  by s

31. General Confidence Interval for μ (Large Samples)
The general formula is:
Where:
 is between a value between 0-1,
(1-)×100% is the confidence level you want
Z1-/2 is a value from the Normal distribution table.
Example: for a 95% CI,  = 0.05
(1-)×100% = 95%
Z1-/2 = 1.96

32. Z-Values
The value of Z1-/2 for other % confidence intervals are given in standard tables.
Confidence Level
α/2
Z1-/2
90%
0.05 (5%)
1.6449
95%
0.025 (2.5%)
1.96
99%
0.005 (0.5%)
2.5758
99.9%
(0.05%)
4.4172

33. Example
Using these we get the following results for the electronic component example:
Note as  gets smaller the CI gets wider
Also, at the same time as n gets bigger the CI narrows – So big samples leads to more precise estimates (i.e. narrower confidence intervals)
Confidence Level
Z1-/2 CI
90%
1.6449
4681 to 7065
95%
1.96
4456 to 7290
99%
2.5758
4011 to 7735
99.9%
4.4172
2679 to 9067

34. What CI’s and sample sizes should I use?
You can’t control s – it is inherent in the data (population).
You can’t control x-bar either.
You can control Z1-/2 but in practice scientific convention sets this to reflect 90%, 95% or 99% confidence, with 95% being the accepted default.
You can choose n – but resources may limit you.
There is a whole topic called sample size determination which you may want to review before collecting data or starting research

35. Confidence Interval Assumptions
Sample size 40 or greater
Experimental units are independent or each other
Experimental units were randomly sampled
The independence assumption requires that value of the variable for one experimental unit should not tell us anything about the value of another.
Randomness is required to avoid systematic bias in selection.

36. Exercise
Complete Exercise 1 & 2

37. Calculation of CIs for small samples
What about small samples?
In the case of CIs about a mean we can use the Student-t distribution.
The process turns of to be very similar – but the CLT no longer works

38. History of the Student t test
William Gosset used the publishing pseudonym ‘Student’. He derived the correct sampling distribution for the mean of samples < 40 – and called it the ‘t distribution’.
In his honour, it is often called the ‘Student t’ distribution.
Gosset was a chief brewer for Guinness.
The mathematical details are complicated, but, it turns out that we perform exactly the same calculations as before, with the one change that the t distribution instead of the normal distribution is used.

39. Assumptions
Student t’s result only referred to a mean where the distribution of the population was normally distributed with some mean μ and finite standard deviation σ.
This is in contrast to the CLT for large samples that required no such assumption about normality.
The t-test also requires the assumption regarding independence in the sample.

40. Statistical Model for mean from small samples
The experimental units are independently sampled from a population with mean=μ and standard deviation = σ
The population is normally distributed (we don’t need this with large samples)
So, to use the t-test for a small sample, you need to establish that data is sampled from a population that is normally distributed – you could look at the histogram of the sample and see if it is symmetric and bell shaped – or use other methods.

41. The t - Statistic If Assumptions met: The statistic:
Can be shown to be distributed according to a (student) t-distribution.
The t-distribution has one parameter, called ‘degrees of freedom’ (df).

42. The t-Distribution
The t-distribution itself is bell shaped and symmetric – just like the normal distribution but is ‘flatter’.
There are many t distributions – one for each sample size.
The rule used is: for a sample of size n – use the t distribution with degrees of freedom = n−1
Example: if the sample size is 15, then use a t distribution with degrees of freedom 15 − 1=14.
Note the degrees of freedom often abbreviated to df.

43. The t-Distribution
The t probability density function with k degrees of freedom:

44. General Confidence Interval for μ (small Samples)
The general formula is:
Where (1-) 100% is the confidence level you want and
t(n-1, /2) is a value from the t distribution with df=n-1, and with a specified  level.
What is t(n−1, 1−/2)?
A value from the t distribution with n−1 df such that
100(1 − )% of values lie within that range around the mean.

45. How do you find t(n−1, 1−/2)?
from a table specifically designed to give it to you or use a computer
Note: as  gets smaller then CI gets wider
as df gets smaller then CI gets wider
Confidence Level
 /2
t(df=1)
t(df=10)
t(df=30)
90%
0.05 (5%)
6.314
1.812
1.697
95%
0.025 (2.5%)
12.71
2.228
2.042
99%
0.005 (0.5%)
63.66
3.169
2.750
99.9%
(0.05%)
636.6
4.587
3.646

46. Example
Internal temperature of autoclaved aerated concrete used in building. An engineer recorded the following data:
23.01, 22.22, 22.04, 22.62, 22.59
95% CI for the population mean?

47. Exercise
Answer Questions 3-6

48. Confidence Intervals for Proportions (Large Samples)
Proportions (including %) are often a statistic of interest
Think of the proportion of defective items on a production line, the proportion of people who respond favourably to a survey question, to proportion of success versus failures in some experiment
Proportions are also covered by the CLT - remember that a proportion is a different kind of average

49. Confidence Intervals for Proportions (Large Samples)
Take a sample of size n of electronic components coming off a production line, a test each one for defects. The statistic of interest is the proportion of defectives produced by the production process.
The estimated proportion from the sample is,
where (p-hat) is the symbol used for the estimated proportion from the sample

50. Confidence Intervals for Proportions (Large Samples)
If the sample size is sufficiently large and we repeat the experiment a large number of times, then:
The sampling distribution of the proportion will be normally distributed by the CLT
The mean of this distribution will be p - i.e. the 'true' population proportion
The standard deviation of the sampling distribution of the proportion, called the standard error of the proportion is estimated by

51. Example:
A pharmaceutical company produces 400,000 capsules per day of a particular drug. They test 200 of the capsules for defects (too much/little active compound). If the population p = 0.05, and they take 10,000 repeated samples this is the histogram they would get

52. Sample Size
How big does the sample have to be for the CLT to work with proportions?
The rule is different than the rule for means. Do the following test.
A rule of thumb: the sample size is big enough if
np > 5 and
n(1-p) > 5

53. General Confidence Interval Formula for a Population Proportion (large Sample)
where  = the confidence level and Z1-/2 = a value from the standard normal distribution such that 100(1-)% of values of a standard normal distribution lie within that range around the mean
So the Z1-/2 values used for a population proportion are the same as those used for a population mean

54. Example
How many voters will give F.F. a first preference in the next general election ? There are 2 different estimates
Researcher A (10 people) => 40%
Researcher B (100 people) => 25%
How much 'better' is estimate B than estimate A ?
Step one: Can we use the formula for large numbers
Researcher A: np = 10 * 0.4 = 4 => 4 is not greater than 5 therefore you cannot used the large number method
Researcher B: np = 100 * 0.25 = 25
n(1-p) = 100 * ( ) = 75
both figures are greater than 5 therefore you can used the large number method

55. Example Continued Researcher B - 95% Confidence Interval
So, the 95% CI is 17% to 33%.

56. Example Continued
NB: If fact we can get a 95% CI for researcher A's findings using small sample theory (exact CI) - this is available in SAS and other software:
Exact CI’s are often based on direct use of probability models.
The method is based directly on calculations for the binomial distribution (see lecture 3)
What do we have to do?
Using the CLT, we found, that the 95% CI was composed of the set of values for the mean, such that an hypothesis test would not reject the null hypotheses for any of those values in the set using the α = 0.05 level.

57. Using SAS we can calculate a 95% CI for Researcher A:
CI 95% for Researcher A = 12% to 74%
which is too wide to be informative anyway!
If we use the same technique for researcher B we get:
CI95 for Researcher B = 17% to 35%
Which is virtually the same as before using the CLT.

58. Exact CI and tests for population proportions
These work for small samples as well as large samples
With large sample will give essentially the same results as CLT
Must be used for small samples, however
Based on the binomial probability distribution.

59. Difference between Exact and CLT based methods
When sample sizes are ‘large’ they will give the same results – but exact tests can be very hard to compute even with modern PCs
When sample sizes are small exact methods must be used
The CIs from small samples tend to be very wide – there is no short cut from collecting as much high quality data as you can manage.

60. Exercise
Answer Question 7-9