Question

The measurements of the bulk modulus of a material at different temperatures is as follows:
T( ͦ C ) 20 500 1000 1200 1400 1500
K (G Pa) 203 197 191 188 186 184
Determine the regression equation for this data.

Answer 1

Answer on Question #62214 – Math – Statistics and Probability

Question

The measurements of the bulk modulus of a material at different temperatures is as follows:

Determine the regression equation for this data.

Solution

We shall derive the linear regression equation, i.e. we must calculate the coefficients $a$ and $b$ in the equation $K = a + bT$. Now we calculate the next values:

$$\bar{T} = \frac{\sum_{i=1}^{6} t_i}{6} = \frac{20 + 500 + 1000 + 1200 + 1400 + 1500}{6} = 936.6667;$$ $$\bar{K} = \frac{\sum_{i=1}^{6} k_i}{6} = \frac{203 + 197 + 191 + 188 + 186 + 184}{6} = 191.50;$$ $$\overline{T K} = \frac{\sum_{i=1}^{6} t_i k_i}{6} = \frac{20 \cdot 203 + 500 \cdot 197 + 1000 \cdot 191 + 1200 \cdot 188 + 1400 \cdot 186 + 1500 \cdot 184}{6} = 175926.67;$$ $$\sigma_T^2 = \frac{1}{6} \sum_{i=1}^{6} (t_i - \bar{T})^2 = \\ = \frac{(20 - 936.67)^2 + (500 - 936.67)^2 + (1000 - 936.67)^2 + (1200 - 936.67)^2 + (1400 - 936.67)^2}{6} + \\ + \frac{(1500 - 936.67)^2}{6} = 272722.22.$$

Now we can obtain the coefficients using the next formulas:

$$b = \frac{\bar{T} \bar{K} - \bar{T} \cdot \bar{K}}{\sigma_T^2} = \frac{175926.67 - 936.6667 \cdot 191.50}{272722.22} = -0.01263;$$ $$a = \bar{K} - b \bar{T} = 191.50 - (-0.01263) \cdot 936.6667 = 203.33.$$

The regression equation for the given data is $K = 203.33 - 0.01 \cdot T$.

Answer: $K = 203.33 - 0.01 \cdot T$.

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