Question

Oysters are categorized for retail as small, medium, or large based on their volume. The grading process is slow and expensive when done by hand. A computer reconstruction of oyster volume based on image analysis would be desirable to speed the process. Engineers designed two programs estimating oyster volume, one based on two‑dimensional ( 2D ) image processing and the other based on three‑dimensional ( 3D ) image processing. We want to know if either approach is a good predictor of actual oyster volume. The results of both programs are given in the table for a sample of 30 oysters. Actual volumes are expressed in cubic centimeters (cm3) , 2D reconstructions in thousands of pixels and 3D reconstruction in millions of volume pixels.

To access the complete data set, click the link for your preferred software format: Excel Minitab JMP SPSS TI R Mac-TXT PC-TXT CSV CrunchIt! Consider the program that estimates oyster volume using 3D digital income processing.

(a) Use the software of your choice to create a scatterplot of 3D volume reconstruction and actual volume, using 3D reconstruction for the explanatory variable. Find the equation of the least‑squares regression line. ?̂ = ?

(b) Select the statement that verifies the conditions for inference.

-The relationship is clearly non‑linear, the scatterplot shows some unusual patterns that would indicate Normally distributed residuals or residuals with a constant standard deviation, and the observations are not independent.

-The relationship is clearly linear, the scatterplot shows some unusual patterns that would result in non‑Normally distributed residuals or residuals without a constant standard deviation, and the observations are independent.

-The relationship is clearly linear, the scatterplot shows no unusual pattern that would indicate not Normally distributed residuals or residuals without a constant standard deviation, and the observations are independent.

-The relationship is non‑linear, the scatterplot shows no unusual pattern that would indicate not Normally distributed residuals or residuals without a constant standard deviation, and the observations are independent.

(c) What are the null and alternative hypotheses to test whether the linear relationship is statistically significant?

-?0:?=0 versus ??:?≠0

-?0:?=0 versus ??:?<0

-?0:?=0 versus ??:?>0

-None of the other choices are correct.

(d) What is the test statistic ? ? (Enter your answer rounded to two decimal places.) ?=

(e) What are the degrees of freedom? (Enter your answer rounded to the nearest whole number.) df=

(f) What is the ? ‑value? You may find Table C helpful.

-?<0.05

-0.05≤?≤0.1

-0.1<?<0.5

-?>0.5

(g) Which is the most appropriate conclusion of your test?

-We have strong evidence of a positive linear relationship between 3D reconstruction and actual oyster volumes.

-None of the other choices are correct.

-We have weak evidence of a positive linear relationship between 3D reconstruction and actual oyster volumes.

-We have no evidence of a positive linear relationship between 3D reconstruction and actual oyster volumes.

The 95% confidence interval for the population slope ? is lower bound to upper bound cm3 per million volume pixels. (Enter your answers rounded to three decimal places.) lower bound:

upper bound:

Data:

Actual	3D	2D
13.04	5.136699	47.907
11.71	4.795151	41.458
17.42	6.453115	60.891
7.23	2.895239	29.949
10.03	3.672746	41.616
15.59	5.72888	48.07
9.94	3.987582	34.717
7.53	2.678423	27.23
12.73	5.481545	52.712
12.66	5.016762	41.5
10.53	3.942783	31.216
10.84	4.052638	41.852
13.12	5.334558	44.608
8.48	3.527926	35.343
14.24	5.679636	47.481
11.11	4.013992	40.976
15.35	5.565995	65.361
15.44	6.303198	50.91
5.67	1.928109	22.895
8.26	3.450164	34.804
10.95	4.707532	37.156
7.97	3.019077	29.07
7.34	2.76816	24.59
13.21	4.945743	48.082
7.83	3.138463	32.118
11.38	4.410797	45.112
11.22	4.558251	37.02
9.25	3.449867	39.333
13.75	5.609681	51.351
14.37	5.292105	53.281

Answer 1

a)

he above scatter plot show's that the actual value is positive correlated with 3D variable and it show any increasing pattern means that they are linearly related each other.

The correlation between between 3D variable and actual value is 0.9765

The equation of the least‑squares regression line.

?̂= ?(b)

?̂=0.41945+2.475*x

Regression Analysis
	r²	0.954	n	30
	r	0.977	k	1
	Std. Error	0.649	Dep. Var.	Actual
ANOVA table
Source	SS	df	MS	F	p-value
Regression	242.8360	1	242.8360	576.93	3.17E-20
Residual	11.7855	28	0.4209
Total	254.6214	29
Regression output				confidence interval
variables	coefficients	std. error	t (df=28)	p-value	95% lower	95% upper
Intercept	0.4196
3D	2.4752	0.1031	24.019	3.17E-20	2.2641	2.6863

c) The null and alternative hypotheses to test whether the linear relationship is statistically significant or not

?0:?=0 versus ??:?≠0

The value of F-stat =576.85 and the value of F-tab= 3.17118E-20

The F-stat value is greater than F-tab value means that Reject Ho

i.e. Conclusion: The linear relationship is statistically significant between 3D variable and Response actual value at 5% of level of significance.(?≠0)

The test statistic ? =0.897872771

The p-value is 3.17118E-20

It is appropriate conclusion given below for p-value with 5% l.o.s.

The above p-value for 3D variable is less than alpha value then Reject Ho (null hypothesis)

It is conclude that the variable is significance at 5% of level of significance.

The 95% confidence interval for the population slope ? is lower bound to upper bound cm3 per million

lower bound =2.264

upper bound =2.6863