The efficiency for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). An article gave the accompanying data on tank temperature (x) and efficiency ratio (y).Temp.174176177178178179180181Ratio0.801.231.460.990.991.160.961.90Temp.184184184184184185185186Ratio1.371.501.672.072.250.821.350.80Temp.186186186188188189190192Ratio1.792.042.741.452.563.101.933.04(a) Determine the equation of the estimated regression line. (Round all numerical values to five decimal places.) (b) Calculate a point estimate for true average efficiency ratio when tank temperature is 186. (Round your answer to four decimal places.) (c) Calculate the values of the residuals from the least squares line for the four observations for which temperature is 186. (Round your answers to four decimal places.)(186, 0.80) (186, 1.79) (186, 2.04) (186, 2.74) Why do they not all have the same sign?These residuals do not all have the same sign because in the case of the second pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were larger than the predicted value.These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were larger than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were smaller than the predicted value.    These residuals do not all have the same sign because in the case of the third pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were smaller than the predicted value.These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were smaller than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were larger than the predicted value.(d) What proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables? (Round your answer to four decimal places.)

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Asked Jul 26, 2019
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The efficiency for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). An article gave the accompanying data on tank temperature (x) and efficiency ratio (y).

Temp. 174 176 177 178 178 179 180 181
Ratio 0.80 1.23 1.46 0.99 0.99 1.16 0.96 1.90

Temp. 184 184 184 184 184 185 185 186
Ratio 1.37 1.50 1.67 2.07 2.25 0.82 1.35 0.80

Temp. 186 186 186 188 188 189 190 192
Ratio 1.79 2.04 2.74 1.45 2.56 3.10 1.93 3.04
(a) Determine the equation of the estimated regression line. (Round all numerical values to five decimal places.)
 

(b) Calculate a point estimate for true average efficiency ratio when tank temperature is 186. (Round your answer to four decimal places.)
 

(c) Calculate the values of the residuals from the least squares line for the four observations for which temperature is 186. (Round your answers to four decimal places.)
(186, 0.80)  
(186, 1.79)  
(186, 2.04)  
(186, 2.74)  

Why do they not all have the same sign?
These residuals do not all have the same sign because in the case of the second pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were larger than the predicted value.These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were larger than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were smaller than the predicted value.    These residuals do not all have the same sign because in the case of the third pair of observations, the observed efficiency ratio was equal to the predicted value. In the cases of the other pairs of observations, the observed efficiency ratios were smaller than the predicted value.These residuals do not all have the same sign because in the cases of the first two pairs of observations, the observed efficiency ratios were smaller than the predicted value. In the cases of the last two pairs of observations, the observed efficiency ratios were larger than the predicted value.

(d) What proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables? (Round your answer to four decimal places.)
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Expert Answer

Step 1

Hi, since the problem posted by you contains multiple sub-parts, we are answering the first three subparts. If you need any specific sub-part to be answered, please submit that particular sub-pat or specify the subpart number in the message box.

Step 2

Part (a): The least squares regression equation:

The least squares regression line can be obtained using EXCEL. Enter the values of x and y in two columns, say in columns A and B in an EXCEL sheet, with the first cell containing the data labels, x and y. Then, follow the procedure given below:

  • Go to Data > Data Analysis > Regression.
  • Enter Input Y Range as $B$2:$B$25 and Input X Range as $A$2:$A$25.
  • Click OK.

The output is as follows:  

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SUMMARY OUTPUT Regression Statistics Multiple R 0.657061828 R Square Adjusted R Square Standard Error 0.4317 0.405899802 0.5345 Observations 24 ANOVA df SS MS Significance F Regression Residual 4.774589335 4.774589 16.71401 0.00048631 22 6.284606498 0.285664 Total 23 11.05919583 Coefficients Standard Error p-value Lower 95% Upper 95% Upper 95.0 % t Stat Lower 95,09% Intercept -16.195 4.370024669 -3.7059 0.001232 -25.2577624 -7.13200944 -25,25776238 -7.132009445 X Variable 1 0.04795777 0.0973 0.023807426 4.088277 0.000486 0.04795777 0.14670493 0.146704928

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Step 3

Now, b is the slope of the regression line, which is approximately 0.0973 and, a is the “Intercept”, which is approximately -16.195.

Therefore,...

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