Question

listed below are the overhead widths (in cm) of seals measured from photographs and the weights (in kg) of the seals. construct a scatterplot, find the value of the linear correlation coefficient r, and find the critical values of r using \\( \alpha = 0.01 \\). is there sufficient evidence to conclude that there is a linear correlation between overhead widths of seals from photographs and the weights of the seals?

\\( \

$$\begin{array} { c | c c c c c c } { \\text { overhead width } } & { 7.1 } & { 7.6 } & { 9.8 } & { 9.4 } & { 8.9 } & { 8.3 } \\\\ \\hline \\text { weight } & { 113 } & { 188 } & { 249 } & { 201 } & { 206 } & { 190 } \\end{array}$$

\\)

click here to view a table of critical values for the correlation coefficient.

construct a scatterplot. choose the correct graph below.

the linear correlation coefficient is \\( r = \\) (round to three decimal places as needed.)

Explanation:

Step1: Define variables

Let \( x = \text{Overhead Width} \), \( y = \text{Weight} \), \( n = 6 \).

Step2: Calculate sums

\( \sum x = 7.1 + 7.6 + 9.8 + 9.4 + 8.9 + 8.3 = 51.1 \)
\( \sum y = 113 + 188 + 249 + 201 + 206 + 190 = 1147 \)
\( \sum xy = (7.1×113)+(7.6×188)+(9.8×249)+(9.4×201)+(8.9×206)+(8.3×190) = 792.3 + 1428.8 + 2440.2 + 1889.4 + 1833.4 + 1577 = 9961.1 \)
\( \sum x^2 = 7.1^2 + 7.6^2 + 9.8^2 + 9.4^2 + 8.9^2 + 8.3^2 = 50.41 + 57.76 + 96.04 + 88.36 + 79.21 + 68.89 = 440.67 \)
\( \sum y^2 = 113^2 + 188^2 + 249^2 + 201^2 + 206^2 + 190^2 = 12769 + 35344 + 62001 + 40401 + 42436 + 36100 = 229051 \)

Step3: Compute r formula numerator

\( n\sum xy - (\sum x)(\sum y) = 6×9961.1 - 51.1×1147 = 59766.6 - 58611.7 = 1154.9 \)

Step4: Compute r formula denominator

\( \sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]} = \sqrt{[6×440.67 - 51.1^2][6×229051 - 1147^2]} = \sqrt{[2644.02 - 2611.21][1374306 - 1315609]} = \sqrt{32.81×58697} ≈ \sqrt{1925858.57} ≈ 1387.75 \)

Step5: Calculate r

\( r = \frac{1154.9}{1387.75} ≈ 0.832 \)

Step6: Find critical values

For \( n=6 \), \( α=0.01 \), critical values \( ±0.917 \).

Step7: Compare r and critical values

\( |0.832| < 0.917 \), so no sufficient evidence.

Answer:

Scatterplot: (Assuming B matches the positive trend; exact graph not visible, but r calculation is 0.832)
Linear correlation coefficient: 0.832
Critical values: ±0.917
Sufficient evidence? No

(Note: Scatterplot choice depends on visual match, but numerical results are as above.)