Download z-Scores and the Normal Curve

Probability
Probability; Sampling Distribution of
Mean, Standard Error of the Mean;
Representativeness of the Sample
Mean
Probability – Frequency View
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Probability is long run relative frequency
Same as relative frequency in the population
Dice toss p(1) = p(2) = …=p(6) = 1/6
Coin flip p(Head) = p(Tail) = .5
Probability & Decision Making
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Decision making like gambling – go with what
is likely.
Lady tasting tea in England. Milk first or
second?
5 cups of tea to taste. What is the probability
she gets it right?
If you cannot tell the difference, how
likely will you be right on all cups?
Cup
Probability
Correct
1
.5
½
2
.25
½*½
3
.125
½*½*½
4
.0625
½*½*½*½
5
.03125
½*½*½*½*½
How many cups would it take to convince you? Convention in social
science is a probability of .05. Using this standard, she would have to get
all 5 right to be convincing in her ability. She did; they were.
Frequency Distribution of the Mean
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What is the distribution of means if we roll
dice once?
What is the distribution of means if we roll
dices twice and take the average?
Three times?
(See Excel File ‘dice’)
Dice
Raw Data
1 Die
M = 3.5
SD = 1.87
Sampling Distributions of Means
Ave of 2 Dice
M = 3.5
SD = 1.23
Ave of 3 Dice
M = 3.5
SD = .99
Notice the mean, standard deviation, and shape of the distributions.
Sampling Distribution
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Notion of trials, experiments, replications
Coin toss example (5 flips, # heads)
Repeated estimation of the mean
Sampling distribution is a distribution of a
statistic (not raw data) over all possible
samples. Same as distribution over infinite
number of trials. Recall dice example.
Estimator
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We use statistics to estimate parameters
Most often X  
Suppose we want to estimate mean height of
students at USF. Sample students, estimate M.
Accuracy of estimate depends mostly upon N
and SD.
Example of Height
Hypothetical data.
RAW DATA
Height of USF Students
  66;   4
Relative Frequency
0.80
Note that graph
shows the population.
0.64
0.48
0.32
0.16
0.00
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82
Heignt in Inches
Raw Data vs. Sampling Distribution
Two Distributions
Raw and Sampling
0.8
Relative Frequency
Means (N=50)
Note middle and
spread of the
two distributions.
How do they
compare?
0.6
0.4
0.2
Raw Data
0.0
50
52
54
56
58
60
62
64
66
68
70
Heignt in Inches
72
74
76
78
80
Definition of Bias
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Statisticians have worked out properties of
sampling distributions
Middle and spread of sampling distribution are
known.
If mean of sampling distribution equals
parameter, statistic is unbiased. (otherwise, it’s
biased.) The sample mean X is unbiased.
Best estimate of  is X .
Definition of Standard Error
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The standard deviation of the sampling
distribution is the standard error. For the
mean, it indicates the average distance of the
statistic from the parameter.
Means (N=50)
Standard error of the mean.
Standard Error
Raw Data
50
52
54
56
58
60
62
64
66
68
70
Heignt in Inches
72
74
76
78
80
Formula: Standard Error of Mean
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X 
To compute the SEM,
use:
X
N
4
X 
 .57
50
For our Example:
Means (N=50)
Standard error = SD of means = .57
Standard Error
Raw Data
50
52
54
56
58
60
62
64
66
68
70
Heignt in Inches
72
74
76
78
80
Review
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What is a sampling distribution?
What is bias?
What is the standard error of a statistic?
Suppose we repeatedly sampled 100 people
at a time instead of 50 for height at USF.
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What would the mean of the sampling
distribution?
What would be the standard deviation of the
sampling distribution?
Definition
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A sampling distribution is a distribution of
_____?
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1 parameters
2 samples
3 statistics
4 variables
Definition
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What is the standard error of the mean?
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1 average distance of standard from the error
2 average distance of raw data (X) from the data
average (X-bar)
3 square root of the sampling distribution of the
variance
4 standard deviation of the sampling distribution of
the mean
Computation
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If the population mean is 50, the population
standard deviation is 2, and the sample size
is 100, what is the standard error of the
mean?
1 .2
2 .5
3 2
4 10
Deciding whether a Sample represents
a Population
Representativeness: degree to which the sample distribution resembles the
population distribution.
z
X 
X
We can use the normal distribution to figure the probability
of a sample mean. If the sample mean is very unlikely
(has a low probability) we conclude the sample does not
represent the population. If it is likely, we conclude it does.
Suppose we grab a sample of 49 students and their mean GPA is 3.7. We
know the population mean is 3.1 and the population SD is .35. Is the
sample representative?
.35 .35
X   3.7  3.2 .5
X 

 .05 z 


 10
7
49
X
.05
.05
Likely?
Probability (Relative Frequency)
z
X 
X
3.9  3.2 .7


 10
.05
.05
Standard Normal Curve
0 .4
50 Percent
Area beyond 10 =?
0 .3
From z table:
34.13 %
0 .2
p = 7.69*10-23
0 .1
13.59%
2.15%
0 .0
-3
-2
-1
0
1
2
Scores in standard deviations from mu
3
Recall that anything
beyond z = 2 is rare;
anything beyond z = 3 is
remote.
Rejection Region
Probability (Relative Frequency)
Place in the curve that is unlikely if the scenario is true. Area totals to probability.
Standard Normal Curve
0 .4
50 Percent
0 .3
34.13 %
0 .2
0 .1
13.59%
2.15%
0 .0
-3
-2
-1
0
1
2
3
Convention is p = .05; That 5
percent of the area least likely to
occur if the scenario is true is the
rejection region. In most cases,
the extremes of both tails are the
places for the rejection region.
The sample is unrepresentative if
it falls far from the center. For z,
the border is +/- 1.96 for p = .05
for 2 tails. For 1 tail, it is 1.65.
Scores in standard deviations from mu
Bottom 2.5 pct
Top 2.5 pct
Review
We know the population mean is 50 and the population standard deviation is
10. We grab 100 people at random and find the mean of the sample is 45.
Does the sample represent the population?