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 (2 D) image processing and the other based on three‑dimensional (3 D) image processing. We want to know if either approach is a good predictor of actual oyster volume. The results of both programs are contained in the table for a sample of 30 oysters. Actual volumes are expressed in cubic centimeters (cm3), 2 D reconstructions in thousands of pixels and 3 D reconstruction in millions of volume pixels.

Actual	2 D	3 D	Actual	2 D	3 D	Actual	2 D	3 D
13.04	47.907	5.136699	10.53	31.216	3.942783	10.95	37.156	4.707532
11.71	41.458	4.795151	10.84	41.852	4.052638	7.97	29.070	3.019077
17.42	60.891	6.453115	13.12	44.608	5.334558	7.34	24.590	2.768160
7.23	29.949	2.895239	8.48	35.343	3.527926	13.21	48.082	4.945743
10.03	41.616	3.672746	14.24	47.481	5.679636	7.83	32.118	3.138463
15.59	48.070	5.728880	11.11	40.976	4.013992	11.38	45.112	4.410797
9.94	34.717	3.987582	15.35	65.361	5.565995	11.22	37.020	4.558251
7.53	27.230	2.678423	15.44	50.910	6.303198	9.25	39.333	3.449867
12.73	52.712	5.481545	5.67	22.895	1.928109	13.75	51.351	5.609681
12.66	41.500	5.016762	8.26	34.804	3.450614	14.37	53.281	5.292105

The table contains a partial Minitab output for prediction when 2 D reconstruction is 50 thousand pixels.

Fit	SE Fit	95% CI	95% PI
13.581	0.285	(12.997 , 14.165 )	(11.088 , 16.074 )

(a) Find a 95% confidence interval, lower bound to upper bound, for the mean actual volume of oysters for which 2 D reconstruction is 50 thousand pixels. (Enter your answers rounded to three decimal places.)

(a) Find a 95% confidence interval, lower bound to upper bound, for the mean actual volume of oysters for which 2 D reconstruction is 50 thousand pixels. (Enter your answers rounded to three decimal places.)

lower bound:

cm3

upper bound:

cm3

(b) Find a 95% confidence interval, lower bound to upper bound, for the actual volume of one randomly selected oyster if that oyster has a 2 D reconstruction of 50 thousand pixels. (Enter your answers rounded to three decimal places.)

lower bound:

cm3

upper bound:

cm3

(c) The purpose of this computer imaging system is to avoid grading oysters by hand. Oysters are labeled as small if their volume is between 0 and 9.9 cm3 , medium if their volume is between 10 and 12.9 cm3 , or large if their volume is greater than 13 cm3.  

Which of the two confidence intervals would you use to assign a grade label to an oyster?

Both confidence intervals are useful.

The 95% confidence interval for the actual volume of one randomly selected oyster.

The 95% confidence interval for the mean actual volume of oysters.

Neither confidence intervals would be useful.

Which statement correctly explain whether the 2 D imaging system is practically useful?

For an oyster with 50 thousand pixels in 2 D reconstruction, we would predict an actual volume between 11.088 and 16.074 cm3 . This is a very large range encompassing both medium or large grades. Therefore, the 2 D model does not seem to be very useful practically.

For an oyster with 50 thousand pixels in 2 D reconstruction, we would predict an actual volume between 12.997 and 14.165 cm3 . This is a reasonable range. Therefore, the 2 D model seems to be useful practically.

For an oyster with 50 thousand pixels in 2 D reconstruction, we would predict an actual volume between 11.088 and 16.074 cm3 . This is a reasonable range. Therefore, the 2 D model does seem to be useful practically.

For an oyster with 50 thousand pixels in 2 D reconstruction, we would predict an actual volume between 12.997 and 14.165 cm3 . This is a very large range encompassing both small or medium grades. Therefore, the 2 D model does not seem to be very useful practically.

Answer 1

(a) Find a 95% confidence interval, lower bound to upper bound, for the mean actual volume of oysters for which 2 D reconstruction is 50 thousand pixels. (Enter your answers rounded to three decimal places.)

lower bound:12.997

upper bound:14.165

this is confidence interval for the conditional mean of response variable and here in the out put is 95%CI= (12.997 , 14.165 )

(b) Find a 95% confidence interval, lower bound to upper bound, for the actual volume of one randomly selected oyster if that oyster has a 2 D reconstruction of 50 thousand pixels. (Enter your answers rounded to three decimal places.)

lower bound: 11.088

upper bound: 16.074

this is prediction interval for and observed value of response variable and here in the out put is 95%PI= (11.088 , 16.074 )

(c) The 95% confidence interval for the mean actual volume of oysters.

Confidence intervals tell you about how well you have determined the mean.

(d) For an oyster with 50 thousand pixels in 2 D reconstruction, we would predict an actual volume between 12.997 and 14.165 cm3 . This is a reasonable range. Therefore, the 2 D model seems to be useful practically.