Question

Physical Characteristics of sharks are of interest to surfers and scuba divers as well as to marine researchers. Because it is difficult to measure jaw width in living sharks, researchers would like to determine whether it is possible to estimate jaw width from body length, which is more easily measured. The following data on x = length (in feet) and y = jaw width (in inches) for 44 sharks was found in various articles appearing in the magazines Skin Diver and Scuba News:

x	y	x	y	x	y	x	y
18.7	17.5	14.6	13.9	16.7	15.2	12.2	14.8
12.3	12.3	15.8	14.7	17.8	18.2	15.2	15.9
18.6	21.8	14.9	15.1	16.2	16.7	14.7	15.3
16.4	17.2	17.6	18.5	12.6	11.6	12.4	11.9
15.7	16.2	12.1	12.0	17.8	17.4	13.2	11.6
18.3	19.9	16.4	13.8	13.8	14.2	15.8	14.3
14.3	13.3	13.6	14.2	16.2	15.7	15.7	14.3
16.6	15.8	15.3	16.9	22.8	21.2	19.7	21.3
9.4	10.2	16.1	16.0	16.8	16.3	18.7	20.8
18.2	19.0	13.5	15.9	13.6	13.0	13.2	12.2
13.2	16.8	19.1	17.9	13.2	13.3	16.8	16.9

Use Minitab to answer the following questions.

Construct scatterplot

Calculate Pearson correlation coefficient.

Determine the equation for the least squares line.

Calculate R2 and interpret

Conduct test of H0: B1 = 0 vs Ha: B1  0 at =.05.

Estimate the mean jaw width for sharks of length 15ft using a 95% confidence interval.

Assess the reasonableness of the assumptions that are required for parts e and f.

Answer 1

a.

b.

Pearson correlation of x and y = 0.875
P-Value = 0.000

c. The least squares line:

d. R²=0.766 i.e. 76.6% of total variation in the sample of y is explained by sample of x through this regression equation.

e. Since P-value corresponding x=0.000<0.05, we reject H₀ at 5% level of significance and conclude that the true slope is significantly different from zero.

f. 95% confidence Estimate of the mean jaw width for sharks of length 15ft: (14.710, 15.569).

g.

From probability plot, we see that the assumption of normality holds. Since points are randomly distributed on both sides of horizontal axis, hence assumptions of linearity and equal variance hold.

Minitab output:

Regression Analysis: y versus x

The regression equation is
y = 0.69 + 0.963 x

Predictor Coef SE Coef T P
Constant 0.688 1.299 0.53 0.599
x 0.96345 0.08228 11.71 0.000

S = 1.37574 R-Sq = 76.6% R-Sq(adj) = 76.0%

Analysis of Variance

Source DF SS MS F P
Regression 1 259.53 259.53 137.12 0.000
Residual Error 42 79.49 1.89
Total 43 339.02

Predicted Values for New Observations

New Obs Fit SE Fit 95% CI 95% PI
1 15.140 0.213 (14.710, 15.569) (12.330, 17.949)

Values of Predictors for New Observations

New Obs x
1 15.0