3.5 ANALYSIS & CONCLUSIONS

Recall: A line of best fit is drawn through a set of data points to best represent a linear relationship between 2 variables Line of best fit is aka Regression Line or Trend

Line Line needs to be closest to each point – the

more varied the position of the points, the harder to draw the line of best fit

CAUSE & EFFECT RELATIONSHIP Observing a relationship between two

variables does not mean that one variable causes a change in the otherOther factors could be involved, or the

correlation could be a coincidence

In this case, it’s not a coincidence than pain accompanies a needle!

HOW TO SHARPEN A PENCIL

Open window (A) and fly kite (B). String (C) lifts small door (D) allowing moths (E) to escape and eat red flannel shirt (F). As weight of shirt becomes less, shoe (G) steps on switch (H) which heats electric iron (I) and burns hole in pants (J). Smoke (K) enters hole in tree (L), smoking out opossum (M) which jumps into basket (N), pulling rope (O) and lifting cage (P), allowing woodpecker (Q) to chew wood from pencil (R), exposing lead. Emergency knife (S) is always handy in case opossum or the woodpecker gets sick and can't work.

CONSIDERING POSSIBLE CAUSE & EFFECT (A)

State whether the claim in each situation is reasonable:A) A scientific study showed a negative

correlation between aerobic exercise and blood pressure. It claimed that the increase in aerobic activity was the cause of the decrease in blood pressure.

CONSIDERING POSSIBLE CAUSE & EFFECT (A)

A) It is reasonable to think there may be a cause and effect relationship. There are many factors that affect blood pressure however. Since this is a scientific study, we might reasonably expect that the researchers made efforts to neutralize other factors, for example, by studying subjects in a very close age and fitness range.

CONSIDERING POSSIBLE CAUSE & EFFECT (B) B) Mila discovered a positive correlation

between gasoline price and average monthly temperature. She concluded that temperature determines the price of gasoline.

CONSIDERING POSSIBLE CAUSE & EFFECT (B)

B) It is not reasonable to say there is a cause-and-effect relationship between temperature and gasoline prices. A more likely explanation for the correlation is that higher temperatures occur in the summer, when more people are travelling out of town for weekends and vacations. This increased demand could cause the price increase.

CONSIDERING POSSIBLE CAUSE & EFFECT (C) Since the 1950s, the concentration of

carbon dioxide in the atmosphere has been increasing. Crime rates in many countries have also increased over this time period. Does more carbon dioxide in the atmosphere cause people to commit crimes?

CONSIDERING POSSIBLE CAUSE & EFFECT (C)

C) It is not reasonable to say there is a cause-and-effect relationship. It is much more likely that carbon dioxide levels and crime rates are each determined by many other factors, such as increasing populations.

CONSIDERING POSSIBLECAUSE & EFFECT You can also look at the linear correlation

coefficient to determine if the two variables in a plot are related by cause-and-effect. If r = , then the two variables are more than

likely related by cause-and-effect If r is close to 0, then the two variables are

most likely not related by cause-and-effect

These relationships can be seen by looking at a graph of data as well If data points are widely scattered, the

variables are most likely not related by cause-and-effect

USING A LINE OF BEST FIT TO MAKE PREDICTIONS

Below are pre-exam term marks and exam marks for some students in a Grade 12 math course.

a) Graph the data and draw the line of best fit.b) Determine the equation of the line of best fit.c) Use the data to predict the exam mark of a

student with a pre-exam term mark of 98%d) Use the data to predict the exam mark of a

student with a pre-exam term mark of 10%

(a)

Equation of a Straight Line

y = mx + b80 = (0.96)(80) + b80 = 76.8 + bb = 80 – 76.8b = 3.2

y = 0.96x + 3.2

(b)

Pre-exam term mark: 98%Exam Mark:

y = 0.96x + 3.2 when x=98y = 0.96(98) + 3.2 = 94.08 + 3.2 = 97.28 or 97%

(c)

Pre-exam term mark: 10%Exam Mark:

y = 0.96x + 3.2 when x=10y = 0.96(10) + 3.2 = 9.6 + 3.2 = 12.8 or 13%

(d)

3.5 HOMEWORK p. 186 #1, 2, 3, 4, 6, 7