To Lecture Notes

LSP 121 -- 9/23/10

Review Questions

1. What is a standard normal histogram?

   Ans: A bell-shaped histogram with center = 0 and spread = 1.

2. What are z and SD+ for the z-scores of a dataset?

   Ans: z = 0, SD+ = 1.

3. What proportion of IQ scores are between 85 and 115 (mean = 100 and SD+ = 15).

   Ans: For 85 and 115, z = -1 and 1. Using the standard normal table, the proportion of scores in (-&infty;, -1] = 0.1587 and the proportion of scores in (&infty;, 1] = 0.8413. Therefore the proportion of scores in (-1, 1] = 0.8413 - 0.1587 = 0.6826 = 68%.

4. What is the 98th percentile for IQ scores?

   Ans: Look up 0.9800 in the body of the standard normal table. the closest entry is 0.9798, for which the corresponding z is 2.25.

Correlation

• The news is filled with examples of correlations and associations:
  Drinking a glass of red wine per day may decrease your chances of a heart attack.

  Taking one aspirin per day may decrease your chances of stroke or of a heart attack.

  Eating lots of certain kinds of fish may improve your health and make you smarter.

  Driving slower reduces your chances of getting killed in a traffic accident.

  Taller people tend to weigh more.

  Pregnant women that smoke tend to have low birthweight babies.

  Animals with large brains tend to be more intelligent.

  The more you study for an exam, the higher the score you are likely to receive.

• The correlation r measures the linear association between two variables.

• The association between two variables is linear if and only if the scatterplot is ellipse-shaped

• The correlation r is always between -1 and 1 inclusive.

• The R-squared value R² is the square of the correlation. It measures the proportion of variation in the dependent variable that can be attributed to the independent variable.

• The R-squared value R² is always between 0 and 1 inclusive.

• Perfect positive linear association. The points are exactly on the trend line.
  Correlation r = 1; R-squared = 1.00

• Large positive linear association. The points are close to the linear trend line.
  Correlation r = 0.9; R=squared = 0.81.

• Small positive linear association. The points are far from the trend line.
  Correlation r = 0.45; R-squared = 0.2025.

• No association. There is no association between the variables.
  Correlation r = 0.0; R-squared = 0.0.

• Small negative association.
  Correlation r = -0.3. R-squared = 0.09.

• Large negative association.
  Correlation r = -0.95; R-squared = 0.9025

• Perfect negative association.
  Correlation r = -1. R-squared = 1.00.

• How high must a correlation be to be considered meaningful? It depends on the discipline. Here are some rough guidelines:

  Discipline	Meaningful If	Meaningful If
  Physics	r < -0.95 or 0.95 < r	0.9 < R2
  Chemistry	r < -0.9 or 0.9 < r	0.8 < R2
  Biology	r < -0.7 or 0.7 < r	0.5 < R2
  Social Sciences	r < -0.6 or 0.6 < r	0.35 < R2

• Caution:   Correlation does not necessarily imply causation.

• If X is correlated with Y, there could be some third variable that causes both.

• Example:   The price of rum in Havana is highly correlated with minister's salaries in New England. Is there causation?

• For a correlation between X and Y to imply causation,
  1. X must precede Y in time,

  2. the causation must be plausable,

  3. common causes from other variables must be ruled out.

• SPSS Analysis:   Download the Excel file Bears85Roster.xls and to the following with SPSS:
  1. Import this dataset into SPSS.

  2. Compute two new variables from existing variables: Meter (Height in Meters) and Kilo (Weight in Kilos).

  3. Compute the correlation of Meter and Kilo.

  4. Draw the scatterplot of Meter (y-variable) vs. Kilo (x-variable) with a linear trend line.

• Caution:   The correlation is misleading if there is a nonlinear relationship between the variables.
  Example 1:   There is a perfect polynomial relationship between X and Y, but the correlation is -0.368.

• Caution:   Outliers can distort the correlation:
  Example 2:   Without the outlier, the correlation is 1; with the outlier the correlation is 0.514.

  Example 3:   Without the outlier, the correlation is 1; with the outlier the correlation is 0.522.

Activity A4

• Finish Activity 3. Work on Activities 4, and 5.