ITED 434 Quality Assurance

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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

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Class Objectives

• Learn about the standard normal distribution
• Discuss descriptive and inferential statistics
• Learn how to calculate proportions under the
  normal curve.
• Discuss sampling distributions
• Learn how to calculate sample size from a normal
  distribution
• Discuss Hypothesis Testing 2 approaches
• Classical method
• P-value method

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Standard normal distribution

• The standard normal distribution is a normal
  distribution with a mean of 0 and a standard
  deviation of 1. Normal distributions can be
  transformed to standard normal distributions by
  the formula
• X is a score from the original normal
  distribution, ? is the mean of the original
  normal distribution, and ? is the standard
  deviation of original normal distribution.

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Standard normal distribution

• A z score always reflects the number of standard
  deviations above or below the mean a particular
  score is.
• For instance, if a person scored a 70 on a test
  with a mean of 50 and a standard deviation of 10,
  then they scored 2 standard deviations above the
  mean. Converting the test scores to z scores, an
  X of 70 would be
• So, a z score of 2 means the original score was 2
  standard deviations above the mean. Note that the
  z distribution will only be a normal distribution
  if the original distribution (X) is normal.

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Applying the formula
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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Area under a portion of the normal curve -
Example 1
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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Example 2
The z table is used to determine that .9772 of
the scores are below a score 2 standard
deviations above the mean.

• Suppose you wanted to know the proportion of
  students receiving scores between 70 and 80. The
  approach is to figure out the proportion of
  students scoring below 80 and the proportion
  below 70.
• The difference between the two proportions is the
  proportion scoring between 70 and 80.
• First, the calculation of the proportion below
  80. Since 80 is 20 points above the mean and the
  standard deviation is 10, 80 is 2 standard
  deviations above the mean.

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Example 2 Contd

• The difference between the two proportions is the
  proportion scoring between 70 and 80.
• Next, calculate the proportion below 70. Note
  that the area of the curve below 70 is 1 standard
  deviation, or .1359
• To calculate the proportion between 70 and 80,
  subtract the proportion above 80 from the
  proportion below 70. That is .8413 - .0228
  .1359.
• Therefore, only 13.59 of the scores are between
  70 and 80.

To calculate the proportion below 70
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Example 3

• Assume a test is normally distributed with a mean
  of 100 and a standard deviation of 15. What
  proportion of the scores would be between 85 and
  105?
• The solution to this problem is similar to the
  solution to the last one. The first step is to
  calculate the proportion of scores below 85.
• Next, calculate the proportion of scores below
  105. Finally, subtract the first result from the
  second to find the proportion scoring between 85
  and 105.

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Example 3
Begin by calculating the proportion below 85. 85
is one standard deviation below the mean
Using the z-table with 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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Example 3
The z-table shows 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.htm
l for Z table.
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Sampling Distributions
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Sampling Distributions

• If you compute the mean of a sample of 10
  numbers, the value you obtain will not equal the
  population mean exactly by chance it will be a
  little bit higher or a little bit lower.
• If you sampled sets of 10 numbers over and over
  again (computing the mean for each set), you
  would find that some sample means come much
  closer to the population mean than others. Some
  would be higher than the population mean and some
  would be lower.
• Imagine sampling 10 numbers and computing the
  mean over and over again, say about 1,000 times,
  and then constructing a relative frequency
  distribution of those 1,000 means.

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5 Samples
15
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
20
10,000 Samples
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Sampling Distributions

• The distribution of means is a very good
  approximation to the sampling distribution of the
  mean.
• The sampling distribution of the mean is a
  theoretical distribution that is approached as
  the number of samples in the relative frequency
  distribution increases.
• With 1,000 samples, the relative frequency
  distribution is quite close with 10,000 it is
  even closer.
• As the number of samples approaches infinity, the
  relative frequency distribution approaches the
  sampling distribution

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Sampling Distributions

• The sampling distribution of the mean for a
  sample size of 10 was just an example there is a
  different sampling distribution for other sample
  sizes.
• Also, keep in mind that the relative frequency
  distribution approaches a sampling distribution
  as the number of samples increases, not as the
  sample size increases since there is a different
  sampling distribution for each sample size.

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Sampling Distributions

• A sampling distribution can also be defined as
  the relative frequency distribution that would be
  obtained if all possible samples of a particular
  sample size were taken.
• For example, the sampling distribution of the
  mean for a sample size of 10 would be constructed
  by computing the mean for each of the possible
  ways in which 10 scores could be sampled from the
  population and creating a relative frequency
  distribution of these means.
• Although these two definitions may seem
  different, they are actually the same Both
  procedures produce exactly the same sampling
  distribution.

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Sampling Distributions

• Statistics other than the mean have sampling
  distributions too. The sampling distribution of
  the median is the distribution that would result
  if the median instead of the mean were computed
  in each sample.
• Students often define "sampling distribution" as
  the sampling distribution of the mean. That is a
  serious mistake.
• Sampling distributions are very important since
  almost all inferential statistics are based on
  sampling distributions.

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Sampling Distribution of the mean

• The sampling distribution of the mean is a very
  important distribution. In later chapters you
  will see that it is used to construct confidence
  intervals for the mean and for significance
  testing.
• Given a population with a mean of ? and a
  standard deviation of ?, the sampling
  distribution of the mean has a mean of ? and a
  standard deviation of s/? N , where N is the
  sample size.
• The standard deviation of the sampling
  distribution of the mean is called the standard
  error of the mean. It is designated by the symbol
  ?.

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Sampling Distribution of the mean

• Note that the spread of the sampling distribution
  of the mean decreases as the sample size
  increases.

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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Spread
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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Standard Error in Relation to Sample Size
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 s, 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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Central Limit Theorem
The central limit theorem states that given a
distribution with a mean µ and variance s2, the
sampling distribution of the mean approaches a
normal distribution with a mean (µ) and a
variance s2/N as N, the sample size, increases.
Go to Central Limit Demonstration http//oak.cats
.ohiou.edu/wallacd1/ssample.html
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Central Limit Theorem

• The central limit theorem also states that the
  larger our set of samples the more normal our
  distribution will be.
• Thus, the sampling distribution of the mean will
  have a normal shape and be come increasingly
  normal in shape as the number of samples
  increases.
• The sampling distribution of the mean will be
  normal regardless of the shape of the population
  distribution.
• Whether the population distribution is
  normal,positively or negatively skewed, unimodal
  or bimodal in shape,the sampling distribution of
  the mean will have a normal shape.

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Central Limit Theorem (Contd)

• In the following example we start out with a
  uniform distribution. The sampling distribution
  of the mean, however, will contain variability in
  the mean values we obtain from sample to sample.
  Thus, the sampling distribution of the mean will
  have a normal shape, even though the population
  distribution does not. Notice that because we are
  taking a sample of values from all parts of the
  population, the mean of the samples will be close
  to the center of the population distribution.

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Hypothesis Testing
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Classical Approach

• The Classical Approach to hypothesis testing is
  to compare a test statistic and a critical value.
  It is best used for distributions which give
  areas and require you to look up the critical
  value (like the Student's t distribution) rather
  than distributions which have you look up a test
  statistic to find an area (like the normal
  distribution).
• The Classical Approach also has three different
  decision rules, depending on whether it is a left
  tail, right tail, or two tail test.
• One problem with the Classical Approach is that
  if a different level of significance is desired,
  a different critical value must be read from the
  table.

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Why not accept the null hypothesis?

• A null hypothesis is not accepted just because it
  is not rejected.
• Data not sufficient to show convincingly that a
  difference between means is not zero do not prove
  that the difference is zero.
• No experiment can distinguish between the case of
  no difference between means and an extremely
  small difference between means.
• If data are consistent with the null hypothesis,
  they are also consistent with other similar
  hypotheses.

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Left Tailed Test H1 parameter lt valueNotice the
inequality points to the left Decision Rule
Reject H0 if t.s. lt c.v.
Right Tailed Test H1 parameter gt valueNotice
the inequality points to the right Decision
Rule Reject H0 if t.s. gt c.v.
Two Tailed Test H1 parameter not equal
valueAnother way to write not equal is lt or
gtNotice the inequality points to both sides
Decision Rule Reject H0 if t.s. lt c.v. (left)
or t.s. gt 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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P-Value Approach

• The P-Value Approach, short for Probability
  Value, approaches hypothesis testing from a
  different manner. Instead of comparing z-scores
  or t-scores as in the classical approach, you're
  comparing probabilities, or areas.
• The level of significance (alpha) is the area in
  the critical region. That is, the area in the
  tails to the right or left of the critical
  values.
• The p-value is the area to the right or left of
  the test statistic. If it is a two tail test,
  then look up the probability in one tail and
  double it.
• If the test statistic is in the critical region,
  then the p-value will be less than the level of
  significance. It does not matter whether it is a
  left tail, right tail, or two tail test. This
  rule always holds.
• Reject the null hypothesis if the p-value is less
  than the level of significance.

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P-Value Approach (Contd)

• You will fail to reject the null hypothesis if
  the p-value is greater than or equal to the level
  of significance.
• The p-value approach is best suited for the
  normal distribution when doing calculations by
  hand. However, many statistical packages will
  give the p-value but not the critical value. This
  is because it is easier for a computer or
  calculator to find the probability than it is to
  find the critical value.
• Another benefit of the p-value is that the
  statistician immediately knows at what level the
  testing becomes significant. That is, a p-value
  of 0.06 would be rejected at an 0.10 level of
  significance, but it would fail to reject at an
  0.05 level of significance. Warning Do not
  decide on the level of significance after
  calculating the test statistic and finding the
  p-value.

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P-Value Approach (Contd)

• Any proportion equivalent to the following
  statement is correct
• 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.
• The null hypothesis is rejected if p is at or
  below the significance level it is not rejected
  if p is above the significance level.
• The degree to which p ends up being above or
  below the significance level does not matter.

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Hypothesis Testing

• Hypothesis testing is a method of inferential
  statistics.
• Researchers very frequently put forward a null
  hypothesis in the hope that they can discredit
  it.
• Data are then collected and the viability of the
  null hypothesis is determined in light of the
  data.
• If the data are very different from what would be
  expected under the assumption that the null
  hypothesis is true, then the null hypothesis is
  rejected.
• If the data are not greatly at variance with what
  would be expected under the assumption that the
  null hypothesis is true, then the null hypothesis
  is not rejected.

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Hypothesis Testing

• Note Failure to reject the null hypothesis is
  not the same thing as accepting the null
  hypothesis.

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Steps to Hypothesis Testing

• 1. The first step in hypothesis testing is to
  specify the null hypothesis (H0) and the
  alternative hypothesis (H1). If the research
  concerns whether one method of presenting
  pictorial stimuli leads to better recognition
  than another, the null hypothesis would most
  likely be that there is no difference between
  methods (H0 µ1 - µ2 0). The alternative
  hypothesis would be H1 µ1 µ2. If the research
  concerned the correlation between grades and
  SAT scores, the null hypothesis would most
  likely be that there is no correlation (H0 ?
  0). The alternative hypothesis would be H1 ?0.
• 2. The next step is to select a significance
  level. Typically the .05 or the .01 level is
  used.
• 3. The third step is to calculate a statistic
  analogous to the parameter specified by the null
  hypothesis. If the null hypothesis were defined
  by the parameter µ1- µ2, then the statistic M1 -
  M2 would be computed.

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Steps to Hypothesis Testing

• 4. The fourth step is to calculate the
  probability value (often called the p value)
  which is the probability of obtaining a
  statistic as different or more different from
  the parameter specified in the null hypothesis
  as the statistic computed from the data. The
  calculations are made assuming that the null
  hypothesis is true. (click here for a concrete
  example)
• 5. The probability value computed in Step 4 is
  compared with the significance level chosen in
  Step 2. If the probability is less than or equal
  to the significance level, then the null
  hypothesis is rejected if the probability is
  greater than the significance level then the
  null hypothesis is not rejected. When the null
  hypothesis is rejected, the outcome is said to
  be "statistically significant" when the null
  hypothesis is not rejected then the outcome is
  said be "not statistically significant."

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Steps to Hypothesis Testing

• 6. If the outcome is statistically significant,
  then the null hypothesis is rejected in favor of
  the alternative hypothesis. If the rejected null
  hypothesis were that µ1- µ2 0, then the
  alternative hypothesis would be that µ1 µ2. If
  M1 were greater than M2 then the researcher would
  naturally conclude that µ1 µ2. (Click here to see
  why you can conclude more than µ1 µ2).
• 7. The final step is to describe the result and
  the statistical conclusion in an understandable
  way. Be sure to present the descriptive
  statistics as well as whether the effect was
  significant or not. For example, a significant
  difference between a group that received a drug
  and a control group might be described as follow
  
• Subjects in the drug group scored significantly
  higher (M 23) than did subjects in the control
  group (M 17), t(18) 2.4, p 0.027.

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Steps to Hypothesis Testing

• The statement that "t(18) 2.4" has to do with
  how the probability value (p) was calculated. A
  small minority of researchers might object to two
  aspects of this wording.
• First, some believe that the significance level
  rather than the probability level should be
  reported. The argument for reporting the
  probability value is presented in another
  section.
• Second, since the alternative hypothesis was
  stated as µ1 µ2, some might argue that it can
  only be concluded that the population means
  differ and not that the population mean for the
  drug group is higher than the population mean for
  the control group.

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Steps to Hypothesis Testing

• This argument is misguided. Intuitively, there
  are strong reasons for inferring that the
  direction of the difference in the population is
  the same as the difference in the sample. There
  is also a more formal argument. A non-significant
  effect might be described as follows
• Although subjects in the drug group scored higher
  (M 23) than did subjects in the control group,
  (M 20), the difference between means was not
  significant, t(18) 1.4, p .179.
• It would not have been correct to say that there
  was no difference between the performance of the
  two groups. There was a difference. It is just
  that the difference was not large enough to rule
  out chance as an explanation of the difference.
  It would also have been incorrect to imply that
  there is no difference in the population. Be sure
  not to accept the null hypothesis.