Statistical Inference'''

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Statistical Inference...

• Sociology 315, Winter 2002
• Week 11 March 18-22

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Definitions Types of Errors
Type I error The null hypothesis is rejected
when it is true. Type II error The null
hypothesis is not rejected when it is
false. There is always a chance of making one of
these errors, but we attempt to minimize the
chance of making errors by setting appropriate
alpha levels
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The lower the set alpha level the greater the
likelihood of Type II error - failing to reject
Ho when it is actually false. The higher the
set alpha level, the greater the chance of Type I
error - rejecting Ho when it is actually true.
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Example Grade inflation?(Situations 1 or 4)
H0 µ 2.7 HA µ gt 2.7
Random sample of students
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Decision Rule Set significance level a 0.05.
If p-value ?0.05, reject null hypothesis.
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If is 2.85
If X is 2.85 and p.07
Mean GPA
Fail to reject null since p-value is greater than
0.05.
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If is 2.95
If X is 2.95 and p.007
Mean GPA
Reject null since p-value is smaller then 0.05.
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If is 3.0
If X is 3.0 and p.002
Mean GPA
Reject null since p-value is smaller then 0.05.
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Overhead Example
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Example (SPSS)
Population of adults
Is average family income 14,680? Or is it higher?
Average family income of 75 sampled adults is
14,960.
Sample of 75 adults
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Example (continued)

• Specify hypotheses.
• H0 ? 14,680
• HA ? gt 14,680 degrees
• Collect data Average family income of 75
  sampled adults is 14, 960. How likely is it that
  a sample of 75 adults would have an average
  family income as high as 14, 960 if the average
  family income was 14, 680?

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Use the sig-value from SPSS to make the decision

• The sig-value represents how likely we would be
  to observe such an extreme sample if the null
  hypothesis were true.
• The sig-value is a probability, so it is a number
  between 0 and 1.
• Close to 0 means unlikely.
• So if sig-value is small, (typically, less than
  0.05), then reject the null hypothesis.

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Example (continued)
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Example (continued)

• The sig-value, 0.698, indicates that, if the
  average family income in the population is 14
  680, it is likely that a sample of 75 adults
  would have - by chance alone - an average income
  of 14 960.
• Decision Fail to reject the null hypothesis.
• Conclude that the average family income is about
  14 680.

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Estimation of Parametres Point Estimation
1. Calculated from a probability sample. 2. The
statistic is an unbiased estimate of the
parametre 3. The estimate is efficient (small
error).
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Example Suppose that we do not know the
population parametres. From a population of
2000, we draw a sample of 100. From this
sample, we determine that x 35 s 12
sx 1.2 sx s 12/10 1.2 ? N What
does a mean of 35 years represent? Does it
approximate the population?
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• 1. What if this was a very unusual sample,
• with a z score of 1.96? (with 2.5 left on
• each tail?) In other words, what if, amongst
• all the sample means we generated, the mean
• of 35 in this sample actually was a z-score
• of 1.96?
• What would ? be?
• We know that ? will be less

Z 1.96
?
X35
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• 2. What if this was a very unusual sample,
• with a z score of -1.96? (with 2.5 left on
• each tail?)
• What would ? be?
• We know that ? will be more

Z -1.96
?
X35
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Where does our sample actually fall relative to
the actual population mean? It could be
anywhere.
Z -1.96
Z 1.96
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What percent of cases fall between z-scores of
2 and 2? 95. In any normal distribution of
sample means, approx. 95 will fall between 2
and 2 stand. deviations. If this sample is one
of the 95 of the random samples of this size
which fall between 1.96 and 1.96 z-score, then
? ? falls somewhere between the two calculated
values of 32.648 and 37.353 years. A sample of
this size is accurate to within 2.352 years, 19
times out of 20 (which is 95).
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p (32.648 years ? ? ? 37.352 years)
.95 The probability that 32.648 is less than or
equal to ? is less than or equal to 37.352
equals .95. The probability that the interval
between 32.648 years and 37.352 years contains ?
is .95. Even if this sample mean is as unusual
as having a z-score of 1.96 or 1.96, ? is
included in the range of values from 32.648 to
37.352. (However, if this sample is more
unusual, i.e. a z-score of 1.98 or more, then
this does not hold true.)
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One-Sample Mean
Population of U. of C. students
Is ?, average number of unwanted pregnancies,
0.02?
How likely would sample have average as extreme
as 0.01 if ? 0.02?
Sample of 1000 students
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One-Sample Mean
H0 ? 0.02 versus HA ? ? 0.02
Calculate z Use z to calculate p-value Reject
if p-value is small.
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Binomial Sampling Distribution
One-Sample Proportion
Population of statistics students
Is p, proportion of students who love their
statistics course, 0.20?
How likely would a sample have a proportion as
extreme as 0.80 if p 0.20?
Sample of 300 students
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One-Sample Proportion
H0 p 0.20 versus HA p ? 0.20
Use sample proportion to calculate P-value.
Reject if P-value is small.