S519: Evaluation of Information Systems

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S519 Evaluation of Information Systems

• Social Statistics
• Chapter 7 Are your curves normal?

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Last week
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This week

• Why understanding probability is important?
• What is normal curve
• How to compute and interpret z scores.

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What is probability?

• The chance of winning a lotter
• The chance to get a head on one flip of a coin
• Determine the degree of confidence to state a
  finding

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Normal curve
Symmetrical (bell-shaped) meanmedianmode Asym
ptotic tail closer to the horizontal axis, but
never touch.
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Normal distribution

• Figure 7.4 P157
• Almost 100 of the scores fall between (-3SD,
  3SD)
• Around 34 of the scores fall between (0, 1SD)

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Normal distribution
The distance between contains Range (if mean100, SD10)
Mean and 1SD 34.13 of all cases 100-110
1SD and 2SD 13.59 of all cases 110-120
2SD and 3SD 2.15 of all cases 120-130
gt3SD 0.13 of all cases gt130
Mean and -1SD 34.13 of all cases 90-100
-1SD and -2SD 13.59 of all cases 80-90
-2SD and -3SD 2.15 of all cases 70-80
lt -3SD 0.13 of all cases lt70
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Z score standard score

• If you want to compare individuals in different
  distributions
• Z scores are comparable because they are
  standardized in units of standard deviations.

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Z score

• Standard score

X the individual score the mean S
standard deviation
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Z score

• Z scores across different distributions are
  comparable
• Z scores represent a distance of z score standard
  deviation from the mean
• Raw score 12.8 (mean12, SD2) ? z0.4
• Raw score 64 (mean58, SD15) ? z0.4

Equal distances from the mean
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Excel for z score

• Standardize(x, mean, standard deviation)
• (a2-average(a2a11))/STDEV(a2a11)

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What z scores represent?

• Raw scores below the mean has negative z scores
• Raw scores above the mean has positive z scores
• Representing the number of standard deviations
  from the mean
• The more extreme the z score, the further it is
  from the mean,

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What z scores represent?

• 84 of all the scores fall below a z score of 1
  (why?)
• 16 of all the scores fall above a z score of 1
  (why?)
• This percentage represents the probability of a
  certain score occurring, or an event happening
• If less than 5, then this event is unlikely to
  happen

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Exercise
Lab

• In a normal distribution with a mean of 100 and a
  standard deviation of 10, what is the probability
  that any one score will be 110 or above?

16 Table B.1 (s-p357)
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If z is not integer
Lab

• Table B.1 (S-P357-358)
• Exercise
• The probability associated with z1.38
• 41.62 of all the cases in the distribution fall
  between mean and 1.38 standard deviation,
• About 92 falls below a 1.38 standard deviation
• How and why?

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Between two z scores

• What is the probability to fall between z score
  of 1.5 and 2.5
• Z1.5, 43.32
• Z2.5, 49.38
• So around 6 of the all the cases of the
  distribution fall between 1.5 and 2.5 standard
  deviation.

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Exercise
Lab

• What is the percentage for data to fall between
  110 and 125 with the distribution of mean100 and
  SD10
• Answer 15.25

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Excel

• NORMSDIST(z)
• To compute the probability associated with a
  particular z score

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Exercise
Lab

• The probability of a particular score occurring
  between a z score of 1 and a z score of 2.5

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What can we do with z score?

• Research hypothesis presents a statement of the
  expected event
• We use statistics to evaluate how likely that
  event is.
• Z tests are reserved for populations
• T tests are reserved for samples

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Exercise
Lab

• Compute the z scores where mean50 and the
  standard deviation 5
• 55
• 50
• 60
• 57.5
• 46

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Exercise
Lab

• Based on a distribution of scores with mean75
  and the standard deviation6.38
• What is the probability of a score falling
  between a raw score of 70 and 80?
• What is the probability of a score falling above
  a raw score of 80?
• What is the probability of a score falling
  between a raw score of 81 and 83?
• What is the probability of a score falling below
  a raw score of 63?