Question

Researchers believe a person’s body is used as a perceptual ruler and people will judge the
size of an object based on its relationship to parts of their body. Specifically, some researchers
thought people with smaller hands will perceive objects to be bigger and hence heavier than
those with larger hands. Linkenauger, Mohler, and Proffitt (2011) collected data on 46 participants,
recording their hand width and estimated weight of bean bags. The results are shown
in Table 10.4.

1. Write the null and alternative hypotheses for this study in words (use the term “association”).

2. Use the applet to determine the least squares regression line for predicting estimated
weight based on hand width. What is the value of the slope of the regression line? What
does this number imply in terms of hand width and estimated weight?

3.You should have found a negative association between hand width and estimated weight
of the bean bag in the sample. The question, however, is if there were no association
between hand width and the weight of an object in the population, how likely is it that
we would get a slope as small (as far below zero) as we did. Let’s apply the 3S strategy.
a. Statistic: What is the value of the slope in the sample?

Answer 1

1. Null hypothesis: The slope of the regression line between width of hands and weight of hands is zero.(i.e alpha is equal to zero where alpha is slope) or There is no linear association between width and weight of hands.

Alternative hypothesis: The slope is not zero (i.e alpha is not equal to zero where alpha is slope) or There is linear association between width and weight of hands.

2. We have , no of observation=46

In our case, The independent variable is Width of hand (X) and Dependent variable is Weight of hand(Y).

mean of x=389.6/46=8.469565

mean of y=4347.9/46=94.519565

Linear regression of Y on X is given by: Y=$\alpha$+$\beta$X where a and b are unknown constants.

Using the least square method of estimation: estimate of $\alpha$ (i.e intercept) and estimate of $\beta$(i.e slope) is

$\alpha$ and $\beta$

$\beta$=Cov(x,y)/Var(x) and $\alpha$=mean of y-$\beta$*mean of x

Cov(x,y)= $\sum (x_{i}*y_{i}})/n-(mean of x* mean of y)$

From above data: Cov(x,y)=36703.28/46-(8.469565*94.519565)=-2.64223

Var(x)=$\sum (x_{i}-mean of x)^{2}/n$=0.719053

b'=-2.64223/0.719053=-3.6746

Similarly from above formula for $\alpha$, $\alpha$=125.6418

So linear regression of Weights of hands on Width of hands is given by,

Y=125.6418-3.6746*X

The slope of the regression line is negative which means there is negative linear relationship between Width of hands and Weight of hands. More precisely, with one unit increase in width of hand, weight of hand decreases by 3.6746 units.

All the formula's used for computations are there in the image attached.

3. (a) Slope of the sample is : -3.6746

The other 2S of the 3S strategy are not mentioned.

We were unable to transcribe this image

We were unable to transcribe this image