ITED 434 Quality Assurance. Statistics Overview: From HyperStat Online Textbook http://davidmlane.com/hyperstat/index.html by David Lane, Ph.D. Rice University. Class Objectives. Learn about the standard normal distribution Discuss descriptive and inferential statistics
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ITED 434Quality Assurance
Statistics Overview: From HyperStat Online Textbook
http://davidmlane.com/hyperstat/index.html
by David Lane, Ph.D. Rice University
ITED 434 - J. Wixson
ITED 434 - J. Wixson
ITED 434 - J. Wixson
Applying the formula will always produce a transformed variable with a mean of zero and a standard deviation of one. However, the shape of the distribution will not be affected by the transformation. If X is not normal then the transformed distribution will not be normal either. One important use of the standard normal distribution is for converting between scores from a normal distribution and percentile ranks.
Areas under portions of the standard normal distribution are shown to the right. About .68 (.34 + .34) of the distribution is between -1 and 1 while about .96 of the distribution is between -2 and 2.
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If a test is normally distributed with a mean of 60 and a standard deviation of 10, what proportion of the scores are above 85?
From the Z table, it is calculated that .9938 of the scores are less than or equal to a score 2.5 standard deviations above the mean. It follows that only 1-.9938 = .0062 of the scores are above a score 2.5 standard deviations above the mean. Therefore, only .0062 of the scores are above 85.
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The z table is used to determine that .9772 of the scores are below a score 2 standard deviations above the mean.
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To calculate the proportion below 70:
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Begin by calculating the proportion below 85. 85 is one standard deviation below the mean:
Using the z-tablewith the value of -1 for z, the area below -1 (or 85 in terms of the raw scores) is .1587.
Do the same for 105
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The z-tableshows that the proportion scoring below .333 (105 in raw scores) is .6304. The difference is .6304 - .1587 = .4714. So .4714 of the scores are between 85 and 105.
Go to:http://davidmlane.com/hyperstat/z_table.htmlfor Z table.
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5 Samples
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10 Samples
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15 Samples
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20 Samples
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100 Samples
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1,000 Samples
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10,000 Samples
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An example of the effect of sample size is shown above. Notice that the mean of the distribution is not affected by sample size.
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A variable's spread is the degree scores on the variable differ from each other.
If every score on the variable were about equal, the variable would have very little spread.
There are many measures of spread. The distributions on the right side of this page have the same mean but differ in spread: The distribution on the bottom is more spread out. Variability and dispersion are synonyms for spread.
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Notice that the graph is consistent with the formulas. If is sm= 10 for a sample size of 1 then sm should be equal to for a sample size of 25. When s is used as an estimate of σ, the estimated standard error of the mean is . The standard error of the mean is used in the computation of confidence intervals and significance tests for the mean.
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60
50
40
95 percent
upper confidence limit
30
20
10
0
60
80
90
100
10
20
30
40
50
70
N
-10
Number of tests
-20
-30
95 percent
lower confidence limit
-40
-50
Figure 11.3
Width of confidence interval versus number of tests.
-60
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SEE TABLE 11.1
Summary of confidence limit formulas
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SEE TABLE 10.6
Summary of common probability distributions.
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The central limit theorem states that given a distribution with a mean μ and variance σ2, the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ2/N as N, the sample size, increases.
Go to Central Limit Demonstration:
http://oak.cats.ohiou.edu/~wallacd1/ssample.html
ITED 434 - J. Wixson
ITED 434 - J. Wixson
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Hypothesis Testing
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Left Tailed Test
H1: parameter < valueNotice the inequality points to the left Decision Rule: Reject H0 if t.s. < c.v.
Right Tailed Test
H1: parameter > valueNotice the inequality points to the right Decision Rule: Reject H0 if t.s. > c.v.
Two Tailed Test
H1: parameter not equal valueAnother way to write not equal is < or >Notice the inequality points to both sides Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right)
The decision rule can be summarized as follows:
Reject H0 if the test statistic falls in the critical region
(Reject H0 if the test statistic is more extreme than the critical value)
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The test statistic is to the p-value as the critical value is to the level of significance and the test is know as a “significance test.”
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