Use the following information to answer problems 1 to 15 below: You have modifie

Question

Use the following information to answer problems 1 to 15 below: You have modified laetisaric acid to 3-hydroxy-acetyl-laetisarate and believe it could be used as an anti-fungal agent to control the growth of fungus on wet corn leaves. You grow the fungus Pythium sp. in B petri dishes containing fungal growth agar (PDA = potato dextrose agar) and various concentrations of your new anti-fungal agent. After 48 hours of growth in an incubator the diameters of all fungal colonies are measured with the following results: Some numbers to help you: sigma x = 140 sigma x^2 = 3,500 sigma y = 239.7 sigma y^2 = 7,353.97 sigma xy = 3,809.5 n = 8 Calculate b_1 Calculate b_0 Set up the ANOVA table for simple linear regression. Calculate the correlation coefficient (Pearson's r) Calculate the Coefficient of Determination AND give an EXACT interpretation in light of these data. Calculate the JOINT 95% confidence intervals for beta_1 and beta_2 Test to see if the y-intercept is zero or not. Use the usual five step procedure at alpha = 0.05 Test to see if the slope is zero or not. Use the usual five step procedure at alpha = 0.05 Calculate the 95% confidence interval for the population correlation coefficient 1.960 for your calculation. Since n is small, use z = 1.960 for your calculation.

Explanation / Answer

The data is first read into R statistical software and the answers are given individually.

> xy <- read.csv("clipboard",sep=" ")
> xylm <- lm(y~x,xy)
> summary(xylm)

Call:
lm(formula = y ~ x, data = xy)

Residuals:
    Min      1Q Median      3Q     Max
-6.7107 -1.4098 0.0875 1.4741 5.8286

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) 33.0500     2.5304 13.061 1.24e-05 ***
x            -0.2479     0.1210 -2.049   0.0864 .
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.92 on 6 degrees of freedom
Multiple R-squared: 0.4116,    Adjusted R-squared: 0.3136
F-statistic: 4.198 on 1 and 6 DF, p-value: 0.08639

1. and 2. Here, b1 = -0.2479 and b0 = 33.0500.

3. The ANOVA table is setup next.

> anova(xylm)
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value Pr(>F)
x          1 64.505 64.505 4.1975 0.08639 .
Residuals 6 92.204 15.367                 
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

4. The correlation coefficient is calculated below and obtained as -0.6416.

> cor(xy$x,xy$y)
[1] -0.641578

5. The coefficient of determination is R-square and from the earlier output it is 0.3136, or equivalently 31.36% of the variation of y-values is explained by the x-values. In terms of the original variables, about 31.36% of the variation of the fungus colony is explained by the 3-hydroxy-acetyl-latisarate.

6. The joint 95% confidence interval is given below:

> confint(xylm)
                 2.5 %      97.5 %
(Intercept) 26.8582731 39.24172688
x           -0.5438783 0.04816402

7. Since 0 does not lie in the 95% confidence interval of the intercept interval, the intercept is significant. However, since 0 lies in the 95% confidence interval of X, the slope is insignificant.

8. The 95% confidence interval for the correlation coefficient contains 0 and hence correlation is insignificant.
> cor.test(xy$x,xy$y)

        Pearson's product-moment correlation

data: xy$x and xy$y
t = -2.0488, df = 6, p-value = 0.08639
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9271045 0.1151583
sample estimates:
      cor
-0.641578

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