Statistical Tests

1
Statistical Tests

• How to tell if something (or somethings) is
  different from something else

2
Populations vs. Samples

• Remember that a population is all the possible
  members of a category that we could measure
• Examples
• the heights of every male or every female
• the temperature on every day since the beginning
  of time
• Ever person who ever has, and ever will, take a
  particular drug

3
Populations vs. Samples

• So a population is kind of abstract - typically
  you couldnt ever hope to measure the entire
  population
• Notable exceptions include
• Standardized tests (mean IQ is 100 with std. dev.
  of 15)
• Special populations such as rare diseases or
  isolated groups of people

4
Populations vs. Samples

• A sample is some subset of a population
• Examples
• The heights of 10 students picked at random
• The participants in a drug trial

5
Populations vs. Samples

• The notation
• Sample statistics are usually regular letters
  like s and
• Population statistics are usually greek letters
  like

? - the population mean
? - the population standard deviation
6
Populations vs. Samples

• Test your intuition
• Under what circumstances does the mean of a
  sample equal the mean of the population from
  which it was drawn?
• What about the standard deviation?
• What if your sample was very small relative to
  the population?

7
Populations vs. Samples

• Test your intuition
• Most importantly What if you took more than one
  sample

8
Central Limit Theorem

• There is a distribution of sample means

9
Central Limit Theorem

• There is a distribution of sample means

The population of IQ scores
100
10
Central Limit Theorem

• There is a distribution of sample means

Your Sample
95
The population of IQ scores
100
11
Central Limit Theorem

• There is a distribution of sample means

Your Sample
103
The population of IQ scores
100
12
Central Limit Theorem

• There is a distribution of sample means

Your Sample
99
The population of IQ scores
100
13
Central Limit Theorem

• There is a distribution of sample means
• This is the sampling distribution of the mean

14
Central Limit Theorem

• What is the mean of the sampling distribution of
  the mean?
• mean of the sampling distribution approaches the
  mean of the population with many resamplings

15
Central Limit Theorem

• What is the standard deviation of the sampling
  distribution of the mean?
• The standard error of the mean

Notice it will always be less than the standard
deviation of the population!
16
Central Limit Theorem

• What is the shape of the sampling distribution of
  the mean?
• Central Limit Theorem the sampling distribution
  of the mean is normal regardless of the shape of
  the underlying distribution !
• This means you can use the Z transform and use
  the Z table

17
The Logic of Statistical Tests
18
Statistical Tests

• Consider a simple example
• you are testing the hypothesis that eating
  walnuts makes people smarter by feeding walnuts
  to a group of 30 subjects and then testing their
  IQ

19
Statistical Tests

• Consider a simple example
• you are testing the hypothesis that eating
  walnuts makes people smarter by feeding walnuts
  to a group of 30 subjects and then testing their
  IQ
• If you are right, then eating walnuts will make
  the average IQ of your subjects be higher than
  the average IQ of all people (the population)
  since, mostly, those other people dont eat
  walnuts much

20
Statistical Tests

• Consider a simple example
• Put another way
• Is this sample (entirely) of walnut eaters
  different from the population of mostly
  non-walnut-eaters

21
Types of Errors

• There are two mistakes you could make

22
Types of Errors

• There are two mistakes you could make
• Type I error or False-Positive - you decide the
  walnut treatment works when it doesnt really
• Type II error or False-Negative - you decide the
  walnuts dont work when really they do

23
Types of Successes

• There are two ways to succeed
• Hit or True-Positive You decide the walnuts do
  make people smarter and, in fact, they really do
• Correct-Rejection or True-Negative You decide
  the walnuts dont work and, in fact they really
  dont

24
Outcome Matrix
Actual Situation
Works Doesnt Work
Works True Positive Type I
Doesnt Work Type II True-Negative
Your Conclusion
25
Statistical Tests

• Consider a simple example
• Your subjects turn out to have a mean IQ of 107.5
  (1/2 S.D. from the mean of the population) after
  eating walnuts

26
Statistical Tests

• What are two reasons why the mean IQ of your
  subjects might be greater than the mean of the
  population?
• you happened to pick 30 very smart people (i.e.
  university students)
• WARNING Type I error is possible!

27
Statistical Tests

• What are two reasons why the mean IQ of your
  subjects might be greater than the mean of the
  population?
• you happened to pick 30 very smart people (i.e.
  university students)
• WARNING Type I error is possible!
• the walnuts worked

28
Statistical Tests

• Usually we are worried about making a type I
  error so we need to know
• What fraction of all possible groups of 30
  subjects would have a mean IQ of 105 or less?

29
Statistical Tests

• Usually we are worried about making a type I
  error so we need to know
• What fraction of all possible groups of 30
  subjects would have a mean IQ of 105 or less?
• In other words, we are interested not in the
  distribution of IQ scores themselves, but rather
  in the distribution of mean IQ scores for groups
  of 30 subjects

30
The Z Test

• as it is more formally known

31
Example Z Test

• Using our example in which we are testing the
  hypothesis that walnuts make people smarter
• null hypothesis is that they dont

107.5
? 100
? 15
32
Example Z Test

• Using our example in which we are testing the
  hypothesis that walnuts make people smarter (null
  hypothesis was that they dont)
• We want to know how many standard errors from the
  mean (of the sampling distribution of means) is
  107.5

107.5
? 15
33
Example Z Test
Heres what weve got
107.5
? 15
n 30
Heres what we can compute
Thats what were after so that we can use the
Z table
34
Example Z Test
Heres what weve got
107.5
? 15
n 30
Heres what we can compute
Which is much less than 15!
35
Example Z Test
Heres what weve got
107.5
? 15
n 30
Heres what we can compute
36
Example Z Test
Heres what weve got
107.5
? 15
n 30
Thus
107.5 isnt half a standard deviation from the
sampling distribution mean!
Its actually more than two and a half standard
deviations from the sampling distribution mean !
37
Example Z Test
Heres what weve got
107.5
? 15
n 30
Looking up 2.739 in the Z table reveals that only
.0031 or .31 of the means in the sampling
distribution of mean IQs (for groups of 30 people
each) would have a mean equal to or greater than
107.5!
38
Example Z Test

• What this means is that you have only a 0.31
  chance of making a type I error if you conclude
  that walnuts made your subjects smarter !

39
Example Z Test

• What this means is that you have only a 0.31
  chance of making a type I error if you conclude
  that walnuts made your subjects smarter !
• Put another way, there is only a 0.31 chance
  that this sample of IQs is taken from the regular
  populationwalnut eaters are different

40
Alpha

• Is .31 small enough? What risk of making a Type
  I error is too great?

41
Alpha

• Is .31 small enough? What risk of making a Type
  I error is too great?
• There is no absolute answer - it depends entirely
  on the circumstances

42
Alpha

• Is .31 small enough? What risk of making a Type
  I error is too great?
• There is no absolute answer - it depends entirely
  on the circumstances
• 5 or probability (p) .05 is generally accepted

43
Alpha

• Is .31 small enough? What risk of making a Type
  I error is too great?
• There is no absolute answer - it depends entirely
  on the circumstances
• 5 or probability (p) .05 is generally accepted
• This rate of making Type I errors (ie. number of
  Type I errors per 100 experiments) is called the
  Alpha Level

44
Statistical Significance

• So we conclude that walnuts have a statistically
  significant effect on IQ with a probability of a
  Type I error of less than 5
• In a research article we might say the effect of
  walnuts on IQ was significant (one-tailed Z test,
  p .0031)