A. Solution Concentration: A chemist studied the concentration of a solution (Y
ID: 3171932 • Letter: A
Question
A. Solution Concentration: A chemist studied the concentration of a solution (Y ) over time (X). Fifteen identical solutions were prepared. The 15 solutions were randomly divided into five sets of three, and the five sets were measured, respectively, after 1, 3, 5, 7, and 9 hours. The results are in a datafile on the course web page.
a) Fit a linear regression function to the data.
b) What is the absolute magnitude of the reduction in the variation of Y when X is introduced into the regression model? What is the relative reduction? What does the relative reduction say about the fitted model?
c) Obtain r and attach the appropriate sign.
d) Test for lack of fit of the regression model for the copier maintenance data. Take = 0.025. State the alternatives, decision rule and conclusion.
3. Refer to the Solution Concentration data in Problem A.
a) Prepare a scatter plot of the data. Based on your plot, does a simple linear regression model appear adequate for modelling the sales data?
b) Use the Box-Cox procedure and standardization discussed in class to find an appropriate power transformation of Y . Evaluate SSE for = 0.2, 0.1, 0, 1, 2. What transformation of Y is suggested?
c) Apply the transformation Y = log_10Y to the data and analyze the data using SLR analysis.
d) Plot the estimated regression line and the transformed data. Does the regression line appear to be a good fit to the transformed data?
e) Obtain the residuals and plot them against the fitted values. Also prepare a Q-Q plot. What do your plots show?
f) Express the estimated regression function for the transformed data in the original units.
Explanation / Answer
A. Solution Concentration: A chemist studied the concentration of a solution (Y ) over time (X). Fifteen identical solutions were prepared. The 15 solutions were randomly divided into five sets of three, and the five sets were measured, respectively, after 1, 3, 5, 7, and 9 hours. The results are in a data file on the course web page.
Solution:
Here, we have to use the regression analysis for the prediction of the concentration of a solution based on the time. For this regression model, the dependent variable Y is given as the concentration of a solution and independent variable X is given a time. The required regression model is given as below:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.900875891
R Square
0.811577371
Adjusted R Square
0.797083323
Standard Error
0.474313539
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
1
12.59712
12.59712
55.9938363
4.6112E-06
Residual
13
2.924653333
0.224973333
Total
14
15.52177333
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
2.575333333
0.248732118
10.35384313
1.2001E-07
2.037980262
3.112686404
X
-0.324
0.043298704
-7.482902933
4.6112E-06
-0.417541163
-0.230458837
a) Fit a linear regression function to the data.
Solution:
The linear regression function to the given data is summarized as below:
Y = 2.5753 – 0.324*X
Concentration = 2.5753 – 0.324*time
b) What is the absolute magnitude of the reduction in the variation of Y when X is introduced into the regression model? What is the relative reduction? What does the relative reduction say about the fitted model?
Solution:
The absolute magnitude or coefficient of determination of the reduction in the variation of Y when X is introduced into the regression model is given as 0.8116, which means about 81.16% of the variation in the dependent variable concentration (Y) is explained by the independent variable time (X).
c) Obtain r and attach the appropriate sign.
Solution:
The value of the correlation coefficient r between the dependent variable concentration (Y) and independent variable time (X) is given as -0.9009, which means there is a strong negative linear association or correlation exists between the dependent variable concentration (Y) and independent variable time (X).
d) Test for lack of fit of the regression model for the copier maintenance data. Take = 0.025. State the alternatives, decision rule and conclusion.
Solution:
Here, we have to test whether there is a statistically significant relationship exists between the dependent variable concentration (Y) and independent variable time (X) or not. For checking this significant relationship we have to use the ANOVA F test. The null and alternative hypothesis for this test is given as below:
Null hypothesis: H0: There is no any statistically significant relationship exists between the dependent variable concentration (Y) and independent variable time (X).
Alternative hypothesis: Ha: There is a statistically significant relationship exists between the dependent variable concentration (Y) and independent variable time (X).
For this we are given level of significance or alpha value as 0.025.
The test statistic F is given as 55.9938363.
The P-value for this test is given as 0.0000046.
P-value < Alpha value
So, we reject the null hypothesis that there is no any statistically significant relationship exists between the dependent variable concentration (Y) and independent variable time (X).
This means we conclude that there is sufficient evidence that there is a statistically significant relationship exists between the dependent variable concentration (Y) and independent variable time (X).
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.900875891
R Square
0.811577371
Adjusted R Square
0.797083323
Standard Error
0.474313539
Observations
15
ANOVA
df
SS
MS
F
Significance F
Regression
1
12.59712
12.59712
55.9938363
4.6112E-06
Residual
13
2.924653333
0.224973333
Total
14
15.52177333
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
2.575333333
0.248732118
10.35384313
1.2001E-07
2.037980262
3.112686404
X
-0.324
0.043298704
-7.482902933
4.6112E-06
-0.417541163
-0.230458837
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