The density of molten salt mixtures, y g/cm3 , was measured at various temperatures x°C. The results were:
xi 250 270 290 310 330 350
yi 1.955 1.935 1.890 1.920 1.895 1.865.
Assume that the graphical checks of problem above are satisfactory.
a. Calculate the estimated variance around the regression line, �)|+ &
b. Is the estimated regression coefficient b, significantly different from zero at the 1% level of significance? Can we conclude that temperature has a significant effect on density in this case?
c. What is the 95% confidence interval for ß, the true slope or regression coefficient?
d. Calculate the 95% confidence interval for the mean value of y at each of x = 250, 300, 350.
e. Calculate the correlation coefficient for this data set.
f. Calculate the coefficient of determination.
Using Excel to fit the regression model
Steps:
1. Click of "data" tab and select "data analysis"
2. From the pop up, select "regression" and click "ok"
3. Under "input_y_range" select the range of cells with y values.
Under "input_x_range" select the range of cells with x values.
4. Check the box for "confidence level" and input 99%
5. Check the box "residuals". Select the output range and click ok
The output from excel is shown below
The estimated model is given as
The residuals are obtained by
Part a)
variance around the regression line is the variance of the residuals which is obtained in from excel as
Part b)
The coefficient
. The test of significance for the coefficient returns a t statistic
and a corresponding
The p value is greater than 0.01 therefore the coefficient "b" is not significant at a 1% significance level. We can conclude that temperature has no significant effect on density
Part c)
The 95% confidence interval for the coefficient is
Part d)
At the confidence level, the t critical value is given as
The standard deviation of the dependent variable y is given by
- When x= 250
The confidence interval is given as
- when x=300
The confidence interval is given as
- when x= 350
Part e)
The correlation coefficient is the square root of the r square value. From the Excel output
Therefore
Part f)
The coefficient of determination is the Value.
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