Construct an ANOVA table and conduct a test for linearity. Use ? = 0.05. Compute
ID: 3133167 • Letter: C
Question
Construct an ANOVA table and conduct a test for linearity. Use ? = 0.05.
Compute and interpret the coefficient of determination.
Construct a 95% confidence interval for the slope of the regression function. Use
it to test for a linear relation between y and x at the 5% level of significance.
Provide a point estimate of the hyperactivity rating when preservative intake is 49
Calculate the corresponding residual.
Obtain a 95% prediction interval for hyperactivity rating when preservative intake
is 49 g. Interpret the interval.
Amount ofSubjective food rating of consumed hyperactivity containing the preservative 36 82 45 49 21 24 58 73 85 52 14 13 14 18Explanation / Answer
Regression Analysis
Solution:
Here, we have to use the regression analysis for finding the linear relationship and regression equation for the purpose of prediction of the subjective rating of hyperactivity based on the amount of the food consumed containing the preservative in gram. The regression analysis for these two variables is summarized as below:
Simple Linear Regression Analysis
Regression Statistics
Multiple R
0.7834
R Square
0.6137
Adjusted R Square
0.5654
Standard Error
3.0448
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
1
117.8331
117.8331
12.7100
0.0073
Residual
8
74.1669
9.2709
Total
9
192.0000
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
1.5257
2.5646
0.5949
0.5683
-4.3883
7.4397
Amount of food consumed containing the preservative
0.1614
0.0453
3.5651
0.0073
0.0570
0.2658
Dependent variable for this study is given as the subjective rating of hyperactivity.
Independent variable for this study is given as the amount of the food consumed containing the preservative.
The correlation coefficient between the two variables such as subjective rating of hyperactivity and the amount of the food consumed containing the preservative in gram is given as 0.7834 which means, there is high positive correlation or strong positive linear association or relationship exists between two variables subjective rating of hyperactivity and the amount of the food consumed containing the preservative. The coefficient of determination or the value of the R square is given as 0.5654 which means about 56.54% of the variation in the dependent variable is explained by the independent variable.
The p-value for this regression analysis is given as 0.0073 which is less than the given level of significance or alpha value 0.05, so we reject the null hypothesis that there is no any significant relationship between the given two variables subjective rating of hyperactivity and the amount of the food consumed containing the preservative.
The residuals for the amount of food consumed is given as below:
Amount of food consumed containing the preservative
Residuals
36
-1.336650083
82
-0.761746821
45
-3.789386401
49
3.564953013
21
0.084577114
24
2.600331675
58
3.112216694
73
-2.309010503
85
2.754007739
52
-3.919292427
The point estimate and confidence interval for the value x = 49 is given as below:
Confidence Interval Estimate
Data
X Value
49
Confidence Level
95%
Intermediate Calculations
Sample Size
10
Degrees of Freedom
8
t Value
2.306004
XBar, Sample Mean of X
52.5
Sum of Squared Differences from XBar
4522.5
Standard Error of the Estimate
3.04481
h Statistic
0.102709
Predicted Y (YHat)
9.435047
For Average Y
Interval Half Width
2.2502
Confidence Interval Lower Limit
7.1848
Confidence Interval Upper Limit
11.68526
For Individual Response Y
Interval Half Width
7.3731
Prediction Interval Lower Limit
2.0619
Prediction Interval Upper Limit
16.80816
Simple Linear Regression Analysis
Regression Statistics
Multiple R
0.7834
R Square
0.6137
Adjusted R Square
0.5654
Standard Error
3.0448
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
1
117.8331
117.8331
12.7100
0.0073
Residual
8
74.1669
9.2709
Total
9
192.0000
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
1.5257
2.5646
0.5949
0.5683
-4.3883
7.4397
Amount of food consumed containing the preservative
0.1614
0.0453
3.5651
0.0073
0.0570
0.2658