BHS 307 Statistics for the Behavioral Sciences

1
BHS 307 Statistics for the Behavioral Sciences

• Chapter 7 Finding Proportions Using the Normal
  Curve

2
Notation

• Sample variance S2
• Population variance s2
• Sample standard deviation S or SD
• Population standard deviation s

3
Interpreting the SD

• For most distributions, the majority of
  observations fall within one standard deviation
  of the mean.
• A very small minority fall outside two standard
  deviations.
• This generalization is true no matter what the
  shape of the distribution.
• It works for skewed distributions.

4
Standard Normal Curve

• By definition has a mean of 0 and an SD of 1.
• Standard normal table gives proportions for
  z-scores using the standard normal curve.
• Proportions on either side of the mean equal .50
  (50) and both sides add up to 1.00 (100).

5
Two Applications

• Finding the proportion of scores above or below a
  specific score.
• What proportion or percent of students scored
  above a certain score on a test?
• Finding a particular score corresponding to a
  given probability or proportion.
• What test score corresponds to the 75th
  percentile on a test?

6
Finding a Proportion

• What is the relative standing of a student with a
  640 on the GRE (mean 500, SD 100)?
• Sketch the problem.
• Plan your solution.
• Convert X to a z-score.
• Find the target area for that z-score in the
  table.

7
Proportions Between Scores

• What proportion of students scored between 400
  and 640 on the GRE?
• Sketch the problem.
• Plan the solution subtract one area from the
  other.
• Convert both numbers to z-scores.
• Find the target area and do the subtraction.

8
Proportions Beyond Two Scores

• How many students got very extreme scores on the
  GRE, above or below 200 points from the mean?
• Draw a sketch.
• Make a plan find the areas in the tails and
  then add them.
• Convert both scores to z-scores.
• Find the target areas and add them.

9
Finding a Specific Score

• If students are graded on a curve, what score is
  the cutoff for an A if you want 20 to earn As?
• Mean 230, SD 5
• Draw a sketch and make a plan.
• Find the proportion (.20) and read what z-score
  correspond to it.
• Convert the z-score back to a test score X m
  (z)(s)

10
Rare and Common Events

• Common events occur in the middle of the
  distribution where the frequency is highest.
• Rare events occur in the tails of the
  distribution where the frequency is very low.
• The middle 95 of scores fall within z-scores of
  1.96 and -1.96.