Scatter Plots: Does Fidgeting Keep you Slim?

Some people don't gain weight even when they overeat. Fidgeting and other
nonexercise activities (NEA) may explain why. Researchers deliberately
overfed 16 healthy young adults for 8 weeks. They measured fat gain and
NEA change. The data is in Table 1.

Table 1

NEA Change (cal) | -94 |-57 |-29 |135 |143 |151 |245 |355 | |Fat Gain (kg)
|4.2 |3.0 |3.7 |2.7 |3.2 |3.6 |2.4 |1.3 | |NEA Change (cal) |392 |473 |486
|535 |571 |580 |620 |690 | |Fat Gain (kg) |3.8 |1.7 |1.6 |2.2 |1.0 |0.4
|2.3 |1.1 | |

Who are the individuals? ____________________________________

What is the explanatory variable? ___________________ Response
Variable? ______________________

We will use the TI Graphing calculator. Enter NEA change data (all 16
values) in L1 and fat gain data (all 16 values) in L2. Turn on PLOT1 and
choose TYPE scatterplot (first selection). XLIST should be explanatory
variable and YLIST should be response variable.

[pic][pic]

Use ZOOMSTAT to GRAPH. Use the scatterplot displayed to answer the

1. What is the type (direction) of the relationship? Positive
Negative

2. What is the form of the relationship? Linear
Nonlinear

3. What is the strength of the relationship? Weak Moderate
Strong

4. Are there any outliers? Yes No If yes, which point(s)
represent possible outliers?

Our eyes can be fooled by how strong a linear relationship is; we need to
use a numerical measure, correlation, or r to accurately describe the
association between NEA and Fat Gain. Correlation measures the strength
and direction (type) of linear relationships.

The formula for correlation is:

Correlation [pic]

This is quite tedious to do by hand, so we will use our calculator to
obtain the value of the correlation.

Using your calculator, press STAT, choose CALC, then 8: LinReg (a+bx).
Enter L1 , L2. Then press the ENTER key to get the correlation

r = _______________

Does this value support your answer for question 3 on page 1? Explain why
or why not.

When a scatterplot shows a linear relationship, we would like to summarize
the overall pattern by drawing a line on the scatterplot. A regression line
describes how a response variable changes as an explanatory variable
changes. The least-squares regression line is the line that makes the sum
of the squares of the vertical distances of the data points to the line as
small as possible. (see figure 15.3 on p315 in text)

The form of the equation of a line is y = a + bx where a is the y-
intercept, the value of y when x=0, and b is the slope, the amount y
changes when x increases by one unit.

Using your calculator, press STAT, choose CALC, 8: LinReg (a+bx). Enter
L1 , L2. Then press the ENTER key to get the coefficients for the least-
squares regression line.

a = _________

b= _________

Equation of the least-squares regression line _______________________

The square of the correlation, r2, is the proportion of the variation in
the values of y that is explained by the linear relationship with x.

r2 = ________

Use the least squares line, y = a + bx , to answer the following questions:

1. If a person had no change in their NEA, what would their weight
gain/loss be? ________________, What point is this on the graph?
_______________________.

2. If a person had 750 cal NEA change, what would their weight gain/loss
be? ___________________

3. What proportion of the variation in fat gain can be explained by the
linear relationship with NEA change? _______________.

4. Should you use this regression line to predict Fat Gain for a NEA
change of 1000 calories? Why or why not?

____________________________________________________________________________
____________________________________________________________________________
________________

Outliers can affect the correlation between two variables. To see how the
correlation can be affected by outliers, remove the person in the list
whose NEA change was 392 calories and Fat Gain 3.8 kg. Now recalculate the
value of the correlation (r).

r = _______________

How did removing this person's data affect the correlation?
____________________________________________________________________________
____________________________________________________________________________
____________________________