Question

The Quantitative Environmental Learning Project reported widths and lengths (in centimeters) of a sample of 88 Puget Sound butter clams. Output is shown for the regression of length on width.

Pearson correlation of Width and Length = 0.989 The regression equation is Length = 0.257 + 1.22 Width
Predictor	Coef	SE Coef	T	P
Constant	0.25689	0.09293	2.76	0.007
Width	1.22013	0.01940	62.89	XXXXX
S = 0.3253     R-Sq = 97.9%     R-Sq(adj) = 97.8%

(a) Why would it be natural for the relationship between length and width to be positive?

A clam should make up in width what it lacks in length.

The wider a clam is, the longer it should be.   

(b) Is the opposite assignment of roles (length is explanatory and width is response) also reasonable?

Yes

No   

(c) If one clam is a centimeter wider than another clam, how much longer do you predict it to be? (Round your answer to two decimal places.)
cm
(d) Which of these is the correct notation for the number 0.257 seen in the regression equation?

ŷr     

μ_y

σ

b₀

s

β₁

b₁

β₀

(e) What length do you predict for a clam that is 4 centimeters wide? (Round your answer to three decimal places.)
cm

(f) What is the typical size of a prediction error for the sample? (Enter your answer to four decimal places.)
  cm
(g) How do we denote typical vertical distance of clam length from the regression line for the population of all butter clams?

b₀

σ     

μ_y

r

β₀

b₁

s

β₁

ŷ

(h) Assume the data to be a representative sample taken from the population of all Puget Sound butter clams, with

b₀, b₁, and s as reported in the output. Which of these do we know exactly for the population?

β₀

β₁     

σ

all of the above

none of the above

(i) What strength does the correlation suggest for the relationship?

very weak

moderately weak     

moderate

moderately strong

very strong

(j) Does the reported value of the correlation tell the strength of the relationship in the sample or in the population?

in the sample

in the population   

(k) How many degrees of freedom hold for performing inference about the slope of the regression line for the larger population of butter clams? (Round your answer to the nearest whole number.)
  degrees of freedom
(l) Suppose that the relationship between length and width of Puget Sound butter clams was not representative of the relationship for the larger population of all butter clams. Which of these would be the case?

The distribution of sample slope b₁ would still be centered at the population slope β₁.

The distribution of standardized sample slope "t" would still be centered at zero.     

both of the above

neither of the above

(m) Suppose we wanted to use the same data set (butter clam widths and lengths) to set up a confidence interval to estimate how much longer than wide butter clams tend to be. What would be the appropriate procedure?

chi-square

paired t     

two-sample t

several-sample F

regression

(n) The P-value has been X-ed out; use the size of the t statistic to get an idea of the size of the P-value, and conclude one of the following.

b₁ ≠ 0

b₁ = 0     

β₁ ≠ 0

β₁ = 0

Answer 1

(a)  A clam should make up in width what it lacks in length.

(b) Yes

(c) 1.22 cm.

(e) 6.357. This should be correct in the screenshot.

(f) 0.0194.

(g) μy​

(h) All of the above

(i) very strong

(l) If the relationship is not representative of the larger population, then the slope would be centered at 0, and no significant relationship exists between length and width of a clam.

Explanation :

(a)  A clam should make up in width what it lacks in length.

(b) Yes

(c) Since the regression equation is Length = 0.257+1.22*Width, it tells us that the slope is 1.22. This means that for every 1 cm increase in clam width, the clam length increases by 1.22 cm.

(e) Use the regression equation and substitute 5 to Width. You have Length = 0.257+1.22*5 = 6.357. This should be correct in the screenshot.

(f) This refers to the standard error of the coefficient or the slope. This value according to the table is 0.0194.

(g) This is μy​ .

(h) All of the above. In a regression model, we have b0​ as the estimator of β0​ , b1​ as the estimator of β1​ , and s as the estimator of σ .

(i) The value of the Pearson correlation coefficient is 0.989 which is very close to 1. This indicates a very strong correlation between length and width of the clam.'

(l) If the relationship is not representative of the larger population, then the slope would be centered at 0, and no significant relationship exists between length and width of a clam.