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.)
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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 |
(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?
(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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