The following data refers to the yield of tomatoes (in kg/plot) for soils with f

Question

The following data refers to the yield of tomatoes (in kg/plot) for soils with four different types of salinity. Salinity is indirectly measured by measuring the electrical conductivity (EC, in units of nmhos/cm) of each plot of land. From the data below, we can see that there were a total of 18 measurements of tomato yield: 6 at EC level of 1.6 nmhos/cm, 4 at EC level of 3.8 nmhos/cm, 4 at EC level of 6.0 nmhos/cm, and 5 at EC level of 10.2 nmhos/cm.  

EC level (nmhos/cm) Tomato Yield (kg/plot) 1.6 59.5, 53.3, 56.8, 63.1, 58.7 3.8 55.2, 59.1, 52.8, 54.5 6.0 51.7, 48.8, 53.9, 49.0 10.2 44.6, 48.5, 41.0, 47.3, 46.1

a) Test the null hypothesis (=0.05) that the population mean tomato yield is the same across the different salinity levels (1.6, 3.8, 6.0, and 10.2 nmhos/cm) versus the alternative that the mean tomato yield is different among at least 2 of the salinity levels.  

Hint: Ignore the actual values of salinity and treat the 4 different salinity levels like 4 different groups. You can use the ANOVA test from chapter 9. Equivalently, you can fit a multiple linear regression model in R with 3 predictors (e.g. an indicator variable for EC level = 3.8, an indicator variable for EC level 6.0, and an indicator level for EC level 10.2) and use the F-test for model utility from chapter 11. Note: you do not need to do both the ANOVA test (from Chapter 9) and the F-test for model utility (from Chapter 11); you can use just one of them to answer the question.

b) Now, fit a simple linear regression model to the 18 observations where y, the dependent variable, is tomato yield, and x, the independent variable, is the actual EC level value (1.6 nmhos/cm, …). Test the null hypothesis (at =0.05) that the coefficient of EC level () is 0 versus the alternative that it is not 0. Interpret your results in the context of the problem.  

c) Compare your results from parts a) and b). Note that in a) you treated salinity as a categorical variable, and in b) you treated salinity as a continuous variable. Recall in the beginning of the quarter when we discussed type of variables (continuous and categorical/discrete), we noted that a variable could sometimes be considered continuous and sometimes categorical/discrete and the decision to treat the variable as continuous or categorical could affect interpretation of results.  

Explanation / Answer

a)

> Chegg=read.csv("Chegg_CSV.csv", sep=",", header=T)
>
> model=lm(Tomato_yield~factor( EC_level), data=Chegg)
> summary(model)

Call:
lm(formula = Tomato_yield ~ factor(EC_level), data = Chegg)

Residuals:
Min 1Q Median 3Q Max
-4.980 -1.758 0.110 1.655 4.820

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 58.280 1.334 43.701 2.27e-16 ***
factor(EC_level)3.8 -2.880 2.000 -1.440 0.17194   
factor(EC_level)6 -7.430 2.000 -3.714 0.00231 **
factor(EC_level)10.2 -12.780 1.886 -6.776 8.93e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.982 on 14 degrees of freedom
Multiple R-squared: 0.7857, Adjusted R-squared: 0.7398
F-statistic: 17.11 on 3 and 14 DF, p-value: 5.875e-05

Conclusion: The estimated p-values of EC level 6 and 10.2 are less than 0.05. Hence, these two EC levels have significant difference Tomato yields from the EC level 1.6 at 0.05 level of significance. Again, we can conclude that the mean tomato yield is different among at least 2 of the salinity levels at 0.05 level of significance.

b) Ans:

> model_1=lm(Tomato_yield~factor( EC_level_1.6), data=Chegg)
> summary(model_1)

Call:
lm(formula = Tomato_yield ~ factor(EC_level_1.6), data = Chegg)

Residuals:
Min 1Q Median 3Q Max
-9.1923 -2.5923 -0.3862 3.4327 8.9077

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 50.192 1.288 38.984 < 2e-16 ***
factor(EC_level_1.6) 1.6 8.088 2.443 3.311 0.00442 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.642 on 16 degrees of freedom
Multiple R-squared: 0.4065, Adjusted R-squared: 0.3695
F-statistic: 10.96 on 1 and 16 DF, p-value: 0.004418

Conclusion: The estimated p-value is 0.00442. Hence, we can not accept the null hypothesis and conclude that the actual slope of EC level 1.6 is not equal to 0 at 0.05 level of significance.

c) Ans: In (a) we compared the Tomato Yield at diffent level of EC whereas, in b) we check whether the EC level 1.6 has the efect on Tomato yields or not.

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