Section 7.3 ~ Best-Fit Lines and Prediction

1
Section 7.3 Best-Fit Lines and Prediction

• Introduction to Probability and Statistics
• Ms. Young

2
Objective
Sec. 7.3

• After this section you will become familiar with
  the concept of a best-fit line for a correlation,
  recognize when such lines have predictive value
  and when they may not, understand how the square
  of the correlation coefficient is related to the
  quality of the fit, and qualitatively understand
  the use of multiple regression.

3
Line of Best-Fit
Sec. 7.3

• The best-fit line (or regression line) on a
  scatterplot is a line that lies closer to the
  data points than any other possible line
• This can be useful to make predictions based on
  existing data
• The line of best-fit should have approximately
  the same number of points above it as it has
  below it and it does not have to start at the
  origin
• The precise line of best-fit can be calculated by
  hand, but is very tedious so often times it is
  estimated by eye or by using a calculator

4
Cautions in Making Predictions from Best-Fit Lines
Sec. 7.3

• Dont expect a best-fit line to give a good
  prediction unless the correlation is strong and
  there are many data points
• If the sample points lie very close to the
  best-fit line, the correlation is very strong and
  the prediction is more likely to be accurate
• If the sample points lie away from the best-fit
  line by substantial amounts, the correlation is
  weak and predictions tend to be much less
  accurate

5
Cautions in Making Predictions from Best-Fit Lines
Sec. 7.3

• Dont use a best-fit line to make predictions
  beyond the bounds of the data points to which the
  line was fit
• Ex. The diagram below represents the
  relationship between candle length and burning
  time. The data that was collected dealt with
  candles that all fall between 2 in. and 4 in.
  Using the line of best fit to make a prediction
  far off from these lengths would most likely be
  inappropriate.
• According to the line of best-fit, a candle with
  a length of 0 in. burns for 2 minutes, an
  impossibility

6
Cautions in Making Predictions from Best-Fit Lines
Sec. 7.3

• A best-fit line based on past data is not
  necessarily valid now and might not result in
  valid predictions of the future
• Ex. Economists studying historical data found
  a strong correlation between unemployment and the
  rate of inflation. According to this
  correlation, inflation should have risen
  dramatically in the recent years when the
  unemployment rate fell below 6. But inflation
  remained low, showing that the correlation from
  old data did not continue to hold.
• Dont make predictions about a population that is
  different from the population from which the
  sample data were drawn
• Ex. you cannot expect that the correlation
  between aspirin consumption and heart attacks in
  an experiment involving only men will also apply
  to women
• Remember that a best-fit line is meaningless when
  there is no significant correlation or when the
  relationship is nonlinear
• Ex. there is no correlation between shoe size
  and IQ, so even though you can draw a line of
  best-fit, it is useless in making any conclusions

7
Example 1
Sec. 7.3

• State whether the prediction (or implied
  prediction) should be trusted in
• each of the following cases, and explain why or
  why not.
• Youve found a best-fit line for a correlation
  between the number of hours per day that people
  exercise and the number of calories they consume
  each day. Youve used this correlation to predict
  that a person who exercises 18 hours per day
  would consume 15,000 calories per day.
• This prediction would be beyond the bounds of the
  data collected and should therefore not be
  trusted
• There is a well-known but weak correlation
  between SAT scores and college grades. You use
  this correlation to predict the college grades of
  your best friend from her SAT scores.
• Since the correlation is weak, that means that
  there is much scatter in the data and you should
  not expect great accuracy in the prediction
• Historical data have shown a strong negative
  correlation between birth rates in Russia and
  affluence. That is, countries with greater
  affluence tend to have lower birth rates. These
  data predict a high birth rate in Russia.
• We cannot automatically assume that the
  historical data still apply today. In fact,
  Russia currently has a very low birth rate,
  despite also having a low level of affluence.

8
Example 1 Contd
Sec. 7.3

• A study in China has discovered correlations that
  are useful in designing museum exhibits that
  Chinese children enjoy. A curator suggests using
  this information to design a new museum exhibit
  for Atlanta-area school children.
• The suggestion to use information from the
  Chinese study for an Atlanta exhibit assumes that
  predictions made from correlations in China also
  apply to Atlanta. However, given the cultural
  differences between China and Atlanta, the
  curators suggestion should not be considered
  without more information to back it up.
• Scientific studies have shown a very strong
  correlation between childrens ingesting of lead
  and mental retardation. Based on this
  correlation, paints containing lead were banned
• Given the strength of the correlation and the
  severity of the consequences, this prediction and
  the ban that followed seem quite reasonable. In
  fact, later studies established lead as an actual
  cause of mental retardation, making the rationale
  behind the ban even stronger.

9
The Correlation Coefficient and Best-Fit Lines
Sec. 7.3

• Recall that the correlation coefficient (r)
  refers to the strength of a correlation
• The correlation coefficient can also be used to
  say something about the validity of predictions
  with best-fit lines
• The coefficient of determination, r², is the
  proportion of the variation in a variable that is
  accounted for by the best-fit line
• Ex. The correlation coefficient for the diamond
  weight and price from the scatterplot on p.307 is
  r 0.777, so r² 0.604. This means that about
  60 of the variation in the diamond prices is
  accounted for by the best-fit line relating
  weight and price and 40 of the variation in
  price must be due to other factors.

10
Example 2
Sec. 7.3

• You are the manager of a large department store.
  Over the years, youve found a reasonably strong
  positive correlation between your September sales
  and the number of employees youll need to hire
  for peak efficiency during the holiday season.
  The correlation coefficient is 0.950. This year
  your September sales are fairly strong. Should
  you start advertising for help based on the
  best-fit line?
• r² 0.903, which means that 90 of the variation
  in the number of peak employees can be accounted
  for by a linear relationship with September
  sales, leaving only 10 unaccounted for
• Because 90 is so high, it is a good idea to
  predict the number of employees youll need using
  the best-fit line

11
Multiple Regression
Sec. 7.3

• Multiple regression is a technique that allows us
  to find a best-fit equation relating one variable
  to more than one other variable
• Ex. Price of diamonds in comparison to carat,
  cut, clarity, and color
• The coefficient of determination (R²) is the most
  common measure in a multiple regression
• This tells us how much of the scatter in the data
  is accounted for by the best-fit equation
• If R²is close to 1, the best-fit equation should
  be very useful for making predictions within the
  range of the data
• If R²is close to 0, the predictions are
  essentially useless