(A) Is there significant evidence for a linear relationship between fungus growt

Question

(A) Is there significant evidence for a linear relationship between fungus growth and acid concentration? Carry out the following tests using = 0.05.

i. First test the hypotheses H0 : = 0 versus HA : 6= 0.

ii. Then test the hypothesis H0 : 1 = 0 versus HA : 1 6= 0.

(B) Compute and interpret a 95% confidence interval for .

(C) Compute and interpret a 95% confidence interval for 1.

(D) It is suggested that acid could be used to retard fungus growth. Could these data be used to verify this claim? If not, what could be said? Briefly explain.

Data Below:

LAETISARIC ACID FUNGUS CONCENTRATION GROWTH X (AG/ml) Y (mm) 33.33 31.0 29.8 27.8 28.0 29.0 25.5 10 23.8 10 18.3 20 15.5 20 11.7 30 10.0 30 11.500 23.644 Sr 10.884 7.850 r 0.9875

Explanation / Answer

Part A.i.

First of all we have to check whether there is significant correlation exists between the given two variables fungus growth and acid concentration or not. For checking this hypothesis we have to use the t test for the population correlation coefficient. The null and alternative hypothesis for this test is given as below:

H0: = 0 versus Ha: 0

We are given

= 0.05

n = 12

df = n – 2 = 12 – 2 = 10

r = -0.9875

The test statistic formula is given as below:

Test statistic = t = r*sqrt[(n – 2)/(1 – r^2)]

t = -0.9875*sqrt((12 – 2)/(1 – (-0.9875)^2))

t = -19.812

P-value = 0.00

P-value <

So, we reject the null hypothesis

So, we conclude that there is sufficient evidence that there is a significant linear relationship or correlation exists between the fungus growth and concentration of acid.

Part A.ii.

Here, we have to use t test for slope which is given as below:

H0: 1 = 0 versus Ha: 1 0

= 0.05

n = 12

df = n – 1 = 12 – 1 = 11

The test statistic formula is given as below:

Test statistic = t = b1/SE(b1)

Here, b1 = r*Sy/Sx = -0.9875*7.850/10.884 = -0.712226663

SE(b1) = 0.035982157

Test statistic = t = b1/SE(b1) = -0.712226663/0.035982157 = -19.7952

P-value = 0.00

= 0.05

P-value <

So, we reject the null hypothesis

So, we conclude that there is sufficient evidence that there is a significant linear relationship or correlation exists between the fungus growth and concentration of acid.

Part B

Confidence interval = r -/+ t*SE

We are given

Confidence level = 95%, df = 10, so critical t = 2.228139

SE = 1/sqrt(n – 3) = 1/sqrt(12 – 3) = 1/3 = 0.33

Lower limit = -0.9875 - 2.228139*0.33 = -1.72278587

Upper limit = -0.9875 + 2.228139*0.33 =-0.25221413

Part c

Confidence interval = b1 -/+ t*SE(b1)

We are given

Confidence interval = 95%, df = 11, so critical t = 2.200985, SE(b1) = 0.035982157

Confidence interval = -0.712226663 -/+ 2.2*0.035982157

Lower limit = -0.712226663 - 2.2*0.035982157 = -0.791387408

Upper limit = -0.712226663 + 2.2*0.035982157 = -0.6330659

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