1. Hypothesis Testing

2. Central Limit Theorem Hypotheses and statistics are
dependent upon this theorem.

3. Central Limit Theorem To understand the Central Limit Theorem
we must understand the difference between
three types of distributions…..

4. A distribution is a collection of data that
is typically displayed in a graph showing the frequency
of outcomes:

5. A distribution of data showing tornado frequency
of outcomes:

7. Of particular interest is the “normal distribution”

8. A normal distribution (bell curve) is the random dispersion of
outcomes found typically with natural events.
It is rich with information

9. Different populations will create differing frequency
distributions, even for the same variable…

10. There are three fundamental types of distributions:
Population distributions

11. There are three types of distributions:
Population distributions

12. There are three types of distributions:
Population distributions
Sample distributions

13. There are three types of distributions:
Population distributions
Sample distributions

14. There are three types of distributions:
Population distributions
Sample distributions
and
3. Sampling distributions

16. A sampling distribution is a
distribution of samples, i.e., a
distribution of statistics taken from
samples.

17. The three distributions are related:
Population distributions
The frequency distributions of a population.

18. Why? The three distributions are related: 2. Sample distributions
The frequency distributions of samples.
The sample distribution should look like
the population distribution…..
Why?

19. The three distributions are related:
2. Sample distributions
The frequency distributions of samples.

20. These three distributions are related:
3. Sampling distributions
The frequency distributions of statistics.

21. Why? The three distributions are related: 2. Sample distributions
The frequency distributions of samples.
The sampling distribution should NOT look like
the population distribution…..
Why?

24. Some questions about this sampling distribution:

25. 1. If the population mean was 40, how many
of the sample means would be larger than 40,
and how many would be less than 40?

26. Regardless of the shape of the distribution
above, the sampling distribution would be
symmetrical around the population mean of 40.

27. 2. What will be the variance of the
sampling distribution?

28. The means of all the samples will be closer
together (have less variance) if the variance of
the population is smaller.

29. The means of all the samples will be closer
together (have less variance) if the size of
each sample (n) gets larger.

31. So the sampling distribution will have a mean
very close to the population mean, and a variance
inversely proportional to the size of the sample (n),
and proportional to the variance of the population.

32. Central Limit Theorem

33. Central Limit Theorem If samples are large, then
the sampling distribution created by those
samples will have a mean equal to the
population mean and a standard deviation
equal to the standard error.

34. Sampling Error = Standard Error

35. The sampling distribution will be a normal curve with:
and

36. This makes inferential statistics possible
because all the characteristics of a normal curve
are known.

37. Hong Kong Temperatures

38. A great example of the theorem in action….
A great example of the theorem in action….

39. Hypothesis Testing A statistic tests a hypothesis: H0
This is called the “null” hypothesis.
It can be seen in a variety of ways:
You idea is not validated.
Nothing happened when the IV was changed.
Nothing is happening!

40. Hypothesis Testing
A statistic tests this hypothesis: H0

41. Hypothesis Testing:
A statistic tests a hypothesis: H0
The alternative or default hypothesis is: HA

42. Hypothesis Testing:
A statistic tests a hypothesis: H0
The alternative or default hypothesis is: HA
A probability is established to test the
“null” hypothesis.

44. Hypothesis Testing: 95% confidence: would mean that there
would need to be 5% or less probability of
getting the null hypothesis;
the null hypothesis
would then be dropped in
favor of the “alternative” hypothesis.

45. Hypothesis Testing:
95% confidence:
1 - confidence level (.95) = alpha

46. The null and the alternative
create two different but
related worlds.

47. But, which one is the
real world??

48. Errors:

49. Errors:

50. Errors:
Type I Error: saying something is
happening when nothing is:
p = alpha
Type II Error: saying nothing is
happening when something is:
p = beta

51. An example from court cases:
An example from court cases:
Just because we can do this with
mathematical precision, it does not
remove the responsibility of thinking!

52. Suppose we flipped a coin
a hundred times….
It came up heads 60 times.
Is it a fair coin?

53. No…. Because a Z-test finds that the probability of doing
that is equal to
We would reject the Null Hypothesis!

54. Suppose we flipped the same
coin a hundred times again…
It came up tails 60 times.
Is it a fair coin? No!

55. Yes…absolutely fair! But we have now thrown the
coin two hundred times, and…
It came up tails 100 times.
Is it a fair coin?
Yes…absolutely fair!

56. Perfectly fair
The probability of rejecting the null hypothesis is ZERO!!

57. Poggendorf figure to one side of the brain or to the other….
Suppose we project a
Poggendorf figure to one side
of the brain or to the other….
and measure error.

58. t(11) = 2.17, p = 0.053 What do you conclude?
Paired Samples Statistics
Mean N Std. Error Mean
Pair 1 Right
Left
t(11) = 2.17, p = 0.053
What do you conclude?

59. Now suppose you did this again with another sample of 12 people.
Paired Samples Statistics
Mean N Std. Error Mean
Pair 1 Right
Left
t(11) = 2.17, p = 0.053
Now suppose you did this again
with another sample of 12 people.
t(11) = 2.10, p = 0.057

60. What do you conclude now?
But the probability of independent events is:
p(A) X p(B) so that:
The Null hypothesis probability for both studies was:
X = 0.003
What do you conclude now?

61. But if the brain
hemispheres are truly
independent….
Then...

62. What do you conclude now?
Paired Samples Statistics
Mean N Std. Error Mean
Pair 1 Right
Left
t(22) = 0.53, p = 0.60
What do you conclude now?