A researcher conduct an experiment to examine the relationship between the weigh
ID: 3154368 • Letter: A
Question
A researcher conduct an experiment to examine the relationship between the weight gain of chicken whose diets had been supplemented by different amounts of the amino acid lysine and the amount of lysine ingested. Because the percentage of lysine in known and we can monitor the amount of feed consumed, we can determine the amount of lysine eaten. A random sample of12 two-week old chickens was selected for the study. Each was caged separately and was allowed to eat at will from feed composed of a base supplemented with lysine. The sample data of weight gains and amounts of lysine eaten over the test period are given below. (y represents with gain in grams, and x represents the amount of lysine ingested in grams).
Chick
y
x
1
14.7
0.09
2
17.8
0.14
3
19.6
0.18
4
18.4
0.15
5
20.5
0.16
6
21.1
0.23
7
17.2
0.11
8
18.7
0.19
9
20.2
0.23
10
16
0.13
11
17.8
0.17
12
19.4
0.21
A. Provide an interpretation of the slope value as it relates to this problem.
C. Calculate a 95% confidence interval for the slope and provide an interpretation of this interval.
D. Use Minitab to find a 95% CI for the mean response of weight gain when x = 0.22 and provide an interpretation for this interval.
E. What would be a proper interpretation of the R-squared value as it relates to this problem?
Chick
y
x
1
14.7
0.09
2
17.8
0.14
3
19.6
0.18
4
18.4
0.15
5
20.5
0.16
6
21.1
0.23
7
17.2
0.11
8
18.7
0.19
9
20.2
0.23
10
16
0.13
11
17.8
0.17
12
19.4
0.21
Explanation / Answer
The regression model is given as below:
Simple Linear Regression Analysis
Regression Statistics
Multiple R
0.8522
R Square
0.7262
Adjusted R Square
0.6988
Standard Error
1.0340
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
1
28.3579
28.3579
26.5221
0.0004
Residual
10
10.6921
1.0692
Total
11
39.0500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
12.5085
1.1917
10.4965
0.0000
9.8533
15.1638
x
35.8280
6.9569
5.1500
0.0004
20.3270
51.3290
A. Provide an interpretation of the slope value as it relates to this problem.
The slope for the regression line is given as 35.8280 which represents the unit change in the variable x as per unit change in the variable y.
B. What is the estimate of the model variance, sigma _{e}^{2} ? Identify the standard error of s(b_{1}) . Use that information to conduct a statistical test of the research hypothesis that for this diet preparation and length of study, there is a direct linear relationship between weight gain and amount of lysince eaten. Use 5% level of significance.
We get the p-value for this regression model as the 0.0004 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 linear relationship exists between the given two variables.
C. Calculate a 95% confidence interval for the slope and provide an interpretation of this interval.
The 95% confidence interval for the slope is given as (20.3270, 51.3290) and we interpret that we are 95% confident that the population slope will be lies between the two values 20.3270 and 51.3290.
D. Use Minitab to find a 95% CI for the mean response of weight gain when x = 0.22 and provide an interpretation for this interval.
The regression equation is
y = 12.5 + 35.8 x
Predictor Coef SE Coef T P
Constant 12.509 1.192 10.50 0.000
x 35.828 6.957 5.15 0.000
S = 1.034 R-Sq = 72.6% R-Sq(adj) = 69.9%
Analysis of Variance
Source DF SS MS F P
Regression 1 28.358 28.358 26.52 0.000
Residual Error 10 10.692 1.069
Total 11 39.050
Unusual Observations
Obs x y Fit SE Fit Residual St Resid
5 0.160 20.500 18.241 0.301 2.259 2.28R
R denotes an observation with a large standardized residual
Predicted Values for New Observations
New Obs Fit SE Fit 95.0% CI 95.0% PI
1 20.391 0.481 ( 19.320, 21.462) ( 17.850, 22.931)
Values of Predictors for New Observations
New Obs x
1 0.220
E. What would be a proper interpretation of the R-squared value as it relates to this problem?
The value for the R square or the coefficient of determination is given as 72.6%. This means about 72.6% of the variation in the dependent variable is explained by the independent variable.
Simple Linear Regression Analysis
Regression Statistics
Multiple R
0.8522
R Square
0.7262
Adjusted R Square
0.6988
Standard Error
1.0340
Observations
12
ANOVA
df
SS
MS
F
Significance F
Regression
1
28.3579
28.3579
26.5221
0.0004
Residual
10
10.6921
1.0692
Total
11
39.0500
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
12.5085
1.1917
10.4965
0.0000
9.8533
15.1638
x
35.8280
6.9569
5.1500
0.0004
20.3270
51.3290
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