Question

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/ft?). An article gave the accompanying data on tank temperature (x) and efficiency ratio (Y). Temp. 173 175 176 177 177 178 179 180 Ratio 0.88 1.35 1.32 1.09 0.99 1.14 1.06 1.70 Temp. 183 183 183 183 183 184 184 185 Ratio 1.47 1.52 1.65 2.19 2.09 0.86 1.33 0.84 Temp. 185 185 185 187 187 188 189 191 Ratio 1.85 1.92 2.62 1.39 2.44 3.08 1.93 3.02 in USE SALT (a) Determine the equation of the estimated regression line. (Round all numerical values to four decimal places.) y (b) Calculate a point estimate for true average efficiency ratio when tank temperature is 185. (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 185. (Round your answers to two decimal places.) (185, 0.84) (185, 1.85) (185, 1.92) (185, 2.62) Why do they not all have the same sign? O 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 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. O 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 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 three decimal places.)

Answer 1

A)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	4380.00	39.73	504.00	9.81	46.62
mean	182.50	1.66	SSxx	SSyy	SSxy

Sample size,   n =   24      
here, x̅ = Σx / n=   182.500          
ȳ = Σy/n =   1.655          
SSxx =    Σ(x-x̅)² =    504.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   46.6      
              
estimated slope , ß1 = SSxy/SSxx =   46.615/504=   0.0925      
intercept,ß0 = y̅-ß1* x̄ =   1.6554- (0.0925 )*182.5=   -15.2240      
              
Regression line is, Ŷ=   -15.2240   + (   0.0925   )*x

............

B)

Predicted Y at X=   185   is              
Ŷ=   -15.22402   +   0.09249   *185=   1.8866   

....

C)

X	Y				PREDICTED Y	RESIDUAL
185	0.84				1.886642	-1.05
185	1.85				1.886642	-0.04
185	1.92				1.886642	0.03
185	2.62				1.886642	0.73

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.

OPTION D

...............

D)

R² =    (SSxy)²/(SSx.SSy) =    0.439
Approximately 0.439 Of variation in observations of variable Y, is explained by variable x

..............

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