Examining Relationships Chapter 7

Suppose we found the age and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the age and weight of these adults? Age 24 30 41 28 50 46 49 35 20 39 Wt 256 124 320 185 158 129 103 196 110 130

Create a scatterplot of the data below. Suppose we found the height and weight of a sample of 10 adults. Create a scatterplot of the data below. Is there any relationship between the height and weight of these adults? Is it positive or negative? Weak or strong? Ht 74 65 77 72 68 60 62 73 61 64 Wt 256 124 320 185 158 129 103 196 110 130

The farther away from a straight line – the weaker the relationship The closer the points in a scatterplot are to a straight line - the stronger the relationship. The farther away from a straight line – the weaker the relationship

Identify as having a positive association, a negative association, or no association. + Heights of mothers & heights of their adult daughters - Age of a car in years and its current value + Weight of a person and calories consumed Height of a person and the person’s birth month NO Number of hours spent in safety training and the number of accidents that occur -

Correlation Coefficient (r)- A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data Pearson’s sample correlation is used most parameter - r (rho) statistic - r

Calculate r. Interpret r in context. Speed Limit (mph) 55 50 45 40 30 20 Avg. # of accidents (weekly) 28 25 21 17 11 6 Calculate r. Interpret r in context. .996 There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

Properties of r (correlation coefficient) legitimate values of r is [-1,1] Strong correlation No Correlation Moderate Correlation Weak correlation

The correlations are the same. value of r does not depend on the unit of measurement for either variable x (in mm) 12 15 21 32 26 19 24 y 4 7 10 14 9 8 12 Find r. Change to cm & find r. The correlations are the same.

value of r does not depend on which of the two variables is labeled x y 4 7 10 14 9 8 12 Switch x & y, then find r. The correlations are the same.

value of r is non-resistant x 12 15 21 32 26 19 24 y 4 7 10 14 9 8 22 Find r. The correlation is not resistant to outliers and is strongly affected by outlying observations

r = 0, but has a definite relationship! value of r is a measure of the extent to which x & y are linearly related A value of r close to zero does not rule out any strong relationship between x and y, just that it is not linear. r = 0, but has a definite relationship!

Association vs. Causation In a famous example of a correlation study, the following results were obtained. Year Number of Methodist Cuban Rum Imported Ministers in New England to Boston (in barrels) ---------------------------------------------------------------------------------------- 1860 63 8376 1865 48 6406 1870 53 7005 1875 64 8486 1880 72 9595 1885 80 10,643 1890 85 11,265 1895 76 10,071 1900 80 10,547 1905 83 11,008 1910 105 13,885 1915 140 18,559 1920 175 23,024 1925 183 24,185 1930 192 25,434 1935 221 29,238 1940 262 34,705

Minister data: (Data on Elmo) r = .9999 So does an increase in ministers cause an increase in consumption of rum? NOOO!!!! That’s a silly conclusion to draw. What’s really going on is that both quantities – Methodist ministers and Cuban rum – were driven upwards by other factors, such as population growth.

Correlation does not imply causation