1. Quantitative Methods Lecture 3
Populations and Samples
Statistics books often assume we already know the true mean or the true variance of the whole population being studied
In the real world, we hardly ever know the true values for the whole population
(If we did, there would be no need to carry out a statistical survey…)
We usually have to estimate the characteristics of the population from sample surveys

2. Estimating the Population Mean
Each time we take a sample and calculate the mean, we are only obtaining information about PART of the TOTAL POPULATION.
We have to use the sample mean (‘x-bar’) as an ESTIMATE of the population mean  (‘mu’) which is usually unknown
As an estimate, x-bar is subject to a margin of error

3. Two more statistics
The Standard Error and the Confidence Interval measure the margins of error on our estimate of the true population mean.
The usual Confidence Interval is x-bar plus or minus approximately two standard errors
(1.96 standard errors to be precise)
In other words, we reckon that our estimated mean is probably within about ± 2 Standard Errors of the true mean
But there are complications…

4. Sampling Distributions enable us to make these estimates
Let’s draw a number of different samples from a normally distributed population
We can calculate the mean of each sample
These sample means give us several different estimates of the true population mean
When plotted, the sample means group fairly tightly round the population mean in a bell-shape which is much narrower than the normal distribution
The larger the samples from which the means are drawn, the tighter this bell-shape will be

5. More on Sampling Distributions
The black curve is a Normal Distribution
The blue curve is a Sampling Distribution of various sample means
If we used larger samples, the means would group more tightly
If we used smaller samples, less tightly

6. The Standard Error of the Mean
It has been found that the Standard Error varies in accuracy with the square root of the number in the sample
So the Standard Error = the Standard Deviation divided by N (“the square root of N”)
Thus for any given Standard Deviation, the larger the N (the number in the sample), the smaller the Standard Error will be.
We use the standard error to estimate the population mean from the sample mean - subject to a margin of error.

7. The 95% Confidence Interval
95% of the Normal Distribution is within ± (plus or minus) 1.96 Standard Deviations of the Mean.
In the same way, probability theory shows that, 95% of the time, the true population mean will lie within ±1.96 Standard Errors of any mean calculated from a large sample.
(Small samples are more complicated!)

8. 95% probability is not certainty
Because we are estimating, we cannot be 100% certain
If something is 95% probable, it is only correct 19 times out of 20
So Confidence Intervals are not infallible, unlike Standard Deviations and Variances
But as long as our samples are large (more than 60) margins of error are fairly small

9. Example A sample of 100 ball-bearings are weighed.
They have a mean weight of 150 grams with a standard deviation of 8 grams.
Find the mean weight of the population as a whole, within the 95% Confidence Interval.
Calculate the Standard Error = Std Deviation / ÖN
= 8 / Ö100 = 8/10 = 0.8
We are 95% certain that the population mean will be within ±1.96*0.8 of the sample mean.
So the population mean will lie between *0.8 and *0.8
i.e. between and

10. Meaning of the Confidence Interval
We call it the 95% Confidence Interval because we are fairly (95%) sure the true mean lies between and
We can choose other Confidence Intervals
If we want to be 99% sure of the true mean, we use a WIDER Confidence Interval of ±2.57 Standard Errors
Then we say we are 99% sure that the true mean lies between 150 ± 2.056

11. Small Samples - A Complication
The smaller the sample, the less accurate the estimate
Instead of using 1.96 times the Standard Error, we have to widen the margin
‘T-tables’ show how much we should widen it
In our example today, N-1 gives the appropriate ‘degrees of freedom’ to be used.
So, if we have a sample of 16 cases, the degrees of freedom = 16-1 = 15
This gives us the row of the table to use

12. See how T-distributions ‘flatten out’
T-distributions change shape by sample size.
The normal distribution is shaped like a bell
The T-distributions are shaped more like a cymbal.
The larger the sample, the more bell-like the T-distribution becomes.

13. T-tables show that for N=16, there are N-1=15 degrees of freedom; so we use 2.13 Standard Errors instead of 1.96 Standard Errors for the 95% CI
T-DISTRIBUTION CRITICAL VALUES
Degrees of P= P=0.01
freedom (for use with (for use with
95% C.I.) 99% C.I.)

14. Large samples reduce margins of error
The smaller the sample, the wider the Confidence Interval becomes in terms of Standard Errors.
But if N is large (at least 60 and preferably more than 120), Standard Errors are reduced (because we divide by a sizeableN)
In addition, we do not have to increase the number of Standard Errors in the Confidence Interval from the basic ±1.96
Taken together, these factors push statisticians towards seeking large samples wherever possible, in order to reduce the margins of error.

15. Inferential Statistics
Putting our Descriptive Statistics to Work

16. Why Inferential Statistics differ from Descriptive Statistics
Means, variances, standard deviations and standard errors are Descriptive Statistics
Give anyone a set of figures and the formula and they should come up with the same answers
Inferential statistics can never tell you if something is true or not
They give you the balance of probabilities about whether something is true.

17. How we make inferences
Provided that the sets of data we are examining are distributed normally (more or less), we can make a number of inferences about how likely (or unlikely) specific events will be
Confidence Intervals are a part of Inferential Statistics - they do not tell us what the population mean IS, only that the population mean is likely to fall between certain limits

18. Inferential Statistics help us to distinguish likely events from unlikely events
Thus it is possible to run statistical tests on measurable samples of data
We select a probability ‘cut-off’ value (e.g. 95% probable versus 5% probable) and make judgements how likely our outcome is
The ‘test statistic’ that we compute tells us whether we have observed a likely event (one that happens 95% of the time) or an unlikely one (one that only happens less than 5% of the time)

19. Null Hypothesis And Alternative Hypothesis
We start with the assumption that nothing is proved - that there is no connection between sets of data, and everything has occurred by chance. This is called the NULL HYPOTHESIS
The ALTERNATIVE HYPOTHESIS is that something unlikely or “significant” links the data
If our test statistic tells us that we have observed an unlikely event, we REJECT the Null Hypothesis and ACCEPT the Alternative Hypothesis

20. Example: the ‘Paired’ T-test
Suppose that we give people a ‘treatment’ (training, or medication, or lessons)
We want to measure whether the ‘treatment’ has improved their results
Provided we can measure the outcome, we can test the same sample of people Before and After Treatment and we use the ‘Paired’ T-test

21. There are many other tests
The paired T-test is a simple test to explain
Others tests we will consider include tests for whether different samples have achieved different mean scores
And tests for whether a score on Variable 1 is linked (‘correlated’) to a score on Variable 2

22. Example: We give people some training and measure how scores differ after it
SCORE BEFORE TRAINING 9 8 6 11 13 16 10
SCORE AFTER TRAINING 12 14 15 10 13 11 9
AFTER minus BEFORE 3 6 7 4 2 -2 -1
PERSON A B C D E F G H I J

23. To calculate our ‘Paired’ T-test
Set up the Null Hypothesis:
Any difference in scores after training has occurred by chance
Set up the Alternative Hypothesis
The difference in mean scores is statistically significant
Choose a decision level (‘alpha’)
Normally 95% vs 5% (or 0.95 vs 0.05)

24. When to reject the Null Hypothesis
If we can show that the probability that the Null Hypothesis is true has dropped BELOW 5%, we can reject the Null Hypothesis
In which case, we accept the Alternative Hypothesis that the training has made a ‘significant’ difference
Otherwise, we accept the Null Hypothesis that the training did not change the mean score

25. Calculating the ‘test statistic’
For each test, we calculate a ‘test statistic’
Then we look in our tables to find out whether that number indicates a likely or an unlikely event
In the case of the Paired T-test, the formula for the test statistic is
(X-m)  Standard Error

26. The T-statistic (or ‘T-ratio’)
In (X-m)  Standard Error
X is the mean difference between before and after scores
m is the expected mean difference between before and after scores assuming the Null Hypothesis is true
Standard Error is the Standard Deviation  ÖN
What will m be?

27. Calculations for our example
SAMPLE MEAN of ‘AFTER minus BEFORE’ column = 23/10 =
STANDARD DEVIATION (calculated in the same manner as last week) =
STANDARD ERROR = STDEV/SQRT(N) = 2.87/(SQRT(10))
T-statistic = (SAMPLE MEAN (2.30) - EXPECTED MEAN (0)) divided by the STD ERROR (0.91)
(2.30-0)/ = T =
Again, why is m 0?

28. What does all this mean?
Now that we have calculated that the T-statistic = 2.53, what happens?
We check this number against the appropriate row of the T-tables
The appropriate row will be N-1, or 9 degrees of freedom
(because N=10)
If our T-statistic is less than the ‘critical value’ in the table, the Null Hypothesis stands
If our T-statistic is greater than the ‘critical value’ in the T-table, the Null Hypothesis falls

29. Bother, there are two columns in the T-tables …
T-DISTRIBUTION CRITICAL VALUES
Degrees of P= P=0.01
freedom (for use with (for use with
95% C.I.) 99% C.I.)
Remember we chose the .95 / .05 cut-off level in advance?
This means we use the left column
Our 2.53 ‘beats’ the Critical Value of 2.26 for 9 degrees of freedom

30. Concluding the test
At our selected probability level, the T-statistic we have calculated is greater than the number in the table
Remind me what this means …
It means that we REJECT the Null Hypothesis
Our result is UNLIKELY to have occurred by chance
We conclude that the training HAS significantly changed the mean score

31. How much have we achieved?
Using probability theory and our test statistic, we have made an assessment of the effectiveness of our training
But note again that 95% significance is not certainty
We are going to be wrong 1 time in 20
In ‘life or death’ situations we may want to be 99% or even 99.9% sure
To be 99% sure, we use the right-hand column in the T-table for our ‘critical value’

32. Plenty to think about … We have covered a lot of ground this week
The Null Hypothesis / Alternative Hypothesis approach is the same for all statistical tests
So is the idea of selecting the acceptable decision level (or ‘alpha’) in advance
But in other tests, we use different statistical calculations and different degrees of freedom to obtain our test statistic

33. And finally:
Suppose we had chosen the 99% / 1% cut-off level for our example, what would the result have been?
(pause for thought)…

34. Happy number-crunching!