Central Limit Theorem, Confidence Intervals, and Basics of Hypothesis Tests

1. Recap of Normal Distribution
Sampling Distribution
Estimation
Steps of Hypothesis Testing

2.  Normal curve: brief review
 “Standardized scores” brief review
 Z-scores
 Transformed scores (e.g., T-score, IQ, SAT)
 Sampling Distributions
 Estimation: Confidence Intervals
 Hypothesis Testing

3. Areas
Z-scores
Transformed scores (e.g., T-scores)

4.  Common shape in nature for many traits
 Symmetrical distribution with one mode
 Contains 100% of all cases in population
 Determined by two parameters:
 Mean (center).
Symbols:  (pop.) and M (sample)
 Standard Deviation (spread)
Symbols:  (pop.) and s (sample)
Variance = 2 (pop) and s2 (sample)

5. X 
X-M
Z-score for individual score:
z 
or

s
Transformed scores with new M, new s:
New X = new M + (z)(new s) (Example: T scores)

6. X X M
Z-score for individual score:
z
s
Transformed scores with new M, new s:
New X = new M + (z)(new s)
(Examples: T scores, IQ, SAT and GRE, etc.)
 
 or



7. Shape of the distribution
Standard Error

8.  No longer dealing with individual scores
 Instead, considering the behavior of a statistic
(computed on samples) relative to parameter
in the population.
 Sampling Distribution = distribution of that
statistic if a very large (infinite) number of
samples are drawn.
 Can be distribution of mean, variance, median.
 Sampling Distribution of the Mean is common.

11. Bias and
Variability
Simulation
from the
Estimation
Chapter

12.  When many samples are drawn, the
distribution of their means is normal.
 This is true regardless of the shape of the
population’s distribution.
 The Mean of the sampling distribution is equal
to the Population Mean 
2 2
 

2 so or
  

N N N M M

13.  Variance of sampling distribution is equal to
population variance divided by N (sample size)
 Standard deviation of the sampling
distribution is called the “Standard Error”
(of the parameter). It is square root of the
variance of the sampling distribution.
2 2
 

2 so or
  

N N N M M

14. Point Estimates
Interval Estimates

15.  Use our knowledge of sampling distribution to
make a guess about the population when we
only have information from a sample.
 First examples are artificially simple: they give
us the population standard deviation while we
don’t know the population mean.

16.  The mean of the distribution of sample means
is equal to the population mean .
 The mean of the sampling distribution is called
the expected value of the statistic
 The sample mean is an unbiased estimator of
the population mean .

17.  The mean of a single sample is an unbiased
estimator of the population mean 
 The Standard Error of the Mean (when we
know the population standard deviation )
gives us information about the expected
deviation score of our sample mean M from
the population parameter 
We use this information, and the areas of the
normal curve, to compute an interval estimate.

18. M = 30,  = 9, N =25, 90% confidence interval
We find z.90 from the Inverse Normal
Calculator (here) for the Unit Normal Curve
(z = 1.6449 for a 90% confidence interval)
 According to the Central Limit theorem, the
middle 90% of sample means will be within
1.6449 M of the population mean 
 Therefore, the probability that the population
mean will be within 1.6449 M is also 90%

19.  An interval around the
sample mean that is
1.6449 * M will contain
 90% of the time.
 Confidence Interval
Simulator lets us see
the effect of different
sample sizes on
intervals for 95% and
99% confidence (here)

20.  Calculate the Standard Error of the Mean:
M = /N = 9/ 25 = 9/5 = 1.80
 Calculate Confidence Interval by hand:
 Lower limit = M – (1.6449*1.80) = 35-2.96082
 Upper limit = M + (1.6449*1.80) = 35+2.96082
 90% Confidence Interval
= [32.03918, 37.96082 ]

21.  Remember:
M is the Standard
Deviation of the
Sampling
Distribution, so we
will use M in the
Inverse Normal
Calculator.

22. Null hypothesis and Alternate Hypotheses
Reject a Null hypothesis when it is very unlikely

23. 1. State a null hypothesis about a population
2. Predict the characteristics of the sample
based on the hypothesis
3. Obtain a random sample from the population
4. Compare the obtained sample data with the
prediction made from the hypothesis
 If consistent, hypothesis is reasonable
 If discrepant, hypothesis is rejected

24. • Trial begins with null hypothesis
(innocent until proven guilty).
• Police and prosecutor gather evidence (data)
about probable innocence.
• If there is sufficient evidence, jury rejects
innocence claim and concludes guilt.
• If there is not enough evidence, jury fails to
convict (but does not conclude defendant
is innocent).

25.  State the hypotheses
 Null hypothesis or H0
 Alternate hypothesis or HA or H1
 Set the criteria for a decision
 Pick an alpha level (.05 is common)
 Collect data and compute sample statistics
 Need mean and standard deviation
 Make a decision

26.  Null hypothesis (H0) states that, in the
general population, there is no change, no
difference, or no relationship
 Alternative hypothesis (H1) states that
there is a change, a difference, or a
relationship in the general population

27.  Distribution of sample means is divided
 Those likely if H0 is true
 Those very unlikely if H0 is true
 Alpha level, or level of significance, is a
probability value used to define “very unlikely”
 Critical region is composed of the extreme
sample values that are very unlikely
 Boundaries of critical region are determined
by alpha level.

30.  Data collected after hypotheses stated
 Data collected after criteria for decision set
 This sequence assures objectivity
 Compute a sample statistic (z-score) to
show the exact position of the sample.

31.  If sample data are in
the critical region,
the null hypothesis is
rejected
 If the sample data are
not in the critical
region, the researcher
fails to reject the null
hypothesis

32.  Hypothesis testing is an inferential process
 Uses limited information to reach a general
conclusion, often leading to action
 Sample data used to draw conclusion
about a population that cannot be
observed.
 Errors are possible and probable

33. Actual Situation
No Effect
H0 True
Effect Exits
H0 False
Experimenter’s
Decision
Reject H0 Type I error Decision correct
Retain H0 Decision correct Type II error

35.  Size of difference between sample mean
and original population mean
 Appears in numerator of the z-score
 Variability of the scores
 Influences size of the standard error
 Sample Size
 Influences size of the standard error

36.  In a two-tailed test, the critical region is
divided on both tails of the distribution.
 Researchers usually have a specific
prediction about the direction of a
treatment effect before they begin.
 In a directional hypothesis or one-tailed
test, the hypotheses specify an increase or
decrease in the population mean

39. It all makes sense when a teacher walks through it.
The ideas become yours when you work with them.

40. Recap of Normal Distribution
Sampling Distribution
Estimation
Steps of Hypothesis Testing