Does Fidgeting Keep You Slim?

Some people don't gain weight even when they overeat.Fidgeting and other non-exercise 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.)

5.0

1.7

1.6

2.2

1.0

0.4

1.8

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 L₁and 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.

Use ZOOMSTAT (ZOOM 9) to GRAPH.Use the scatterplot displayed to answer the following questions.Circle your answers:

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

2.What is the form of the relationship?LinearNonlinear

3.What is the strength of the relationship?WeakModerateStrong

4.Are there any outliers?YesNoIf 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, orrto accurately describe the association between NEA and Fat Gain.Correlation measures thestrengthanddirection (type)of linear relationships.

The formula for correlation is:

Correlation

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. Aregression linedescribes how a response variable changes as an explanatory variable changes.Theleast-squares regression lineis 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 p339 in text)

The form of the equation of a line is y = a + bx whereais they-intercept,the value of y when x=0,andbis theslope, 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 =

The equation of the least-squares regression line is:

Thesquare 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 fat gain/loss be?

What is this point called on the graph of the least-squares regression line?

2.If a person had 500 calorie NEA change, what would their fat 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?

5.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 5.0 kg. Now recalculate the value of the correlation (r).

r =

How did removing this person's data affect the strength of the correlation?