Question

Please solve USING RSTUDIO !!

1. You are interested in hormone metabolism in plants. So, you inject 10 mg of gibberellin into 10 plants, then measure the amount (mmol/g) of the hormone evident in tissues on the ten days following injection.

a. Examine the technical assumptions of regression (regardless, proceed with linear regression).

b. Examine the influence (leverage) of the data points in this data set (regardless, do not omit any points).

c. Is there a significant change in gibberellin concentration over time?

d. Estimate and provide the parameters of the best-fit (ordinary least squares) linear model.

e. Predict, with confidence intervals, the gibberellin content of tissues at 1.5 days and 8.5 days.

2. In ring-necked pheasants, both sexes have long, pointy tails (though males have longer, pointier ones). A study wanted to look at the importance of this trait in social pair formation. So, the length of the tail (cm) was measured in 20 female and 20 male pheasants that were in pair bonds. The data are below. Are these characters correlated? Explain.

Day Gibberellin 26.0 20.5 22.5 16.6 14.7 10.4 14.0 10.5 9.6 2.3 2 7 10 Pair Male Female 26 10 2 3 26 23 26 20 28 27 31 10 27 27 12 25 13 27 14 29 15 3:2 13 14 19 15 15 18 20 21 20 21 23 17 1636 1737 18 19 38 2038 18 19 19 38 28

Answer 1

1.

y <- c(26,20.5,22.5,16.6,14.7,10.4,14,10.5,9.6,2.3)
x <- c(1:10)
plot(x,y, col = "red", main = "Gibberellin vs Day")

Gibberellin vs Day CN 2 4 6 8 10

Linear_Reg = lm(formula = y ~ x )
summary(Linear_Reg)

Output:

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.2139 -1.7614  0.0821  2.0886  2.5782 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  26.7667     1.6332  16.389 1.94e-07 ***
x            -2.1921     0.2632  -8.328 3.26e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.391 on 8 degrees of freedom
Multiple R-squared:  0.8966,    Adjusted R-squared:  0.8837 
F-statistic: 69.36 on 1 and 8 DF,  p-value: 3.265e-05

y = 26.76 - 2.19*x    // Predicted line

a. The assumptions for linear regression are:

• Linearity: The relationship between X and the mean of Y is linear.
• Homoscedasticity: The variance of residual is the same for any value of X.
• Independence: Observations are independent of each other.
• Normality: For any fixed value of X, Y is normally distributed.

b. All the data are used in linear regression no any outlier

c. y is continuously decreases wrt to days

d.    Intercept - 26.7667 Slope - 2.1921

e.    y = 26.76 - 2.19*x  

       x = 1.5, y = 23.475

       x = 8.5, y = 8.145

>> abline(Linear_Reg,cex = 1.3,col = "green" )

Gibberellin vs Day CN 2 4 6 8 10