Question

3. Suppose that we had the following results from an experiment in which we measured the growth of a cell culture (as optical density) at different pH levels.

pH 4 5 5.5 6 6.5 7

Optical Density 0.15 0.22 0.26 0.31 0.35 0.41

(a) Find the equation that fits these data.

(b) Is there a positive correlation between pH levels and Optical Density? Test at the 5% level of significance.

please solve the question as soon as possible i need it before 8.00pm today

Answer 1

X	Y	XY	X²	Y²
4	0.15	0.6	16	0.0225
5	0.22	1.1	25	0.0484
5.5	0.26	1.43	30.25	0.0676
6	0.31	1.86	36	0.0961
6.5	0.35	2.275	42.25	0.1225
7	0.41	2.87	49	0.1681

Ʃx =	34
Ʃy =	1.7
Ʃxy =	10.135
Ʃx² =	198.5
Ʃy² =	0.5252
Sample size, n =	6
x̅ = Ʃx/n = 34/6 =	5.666666667
y̅ = Ʃy/n = 1.7/6 =	0.283333333
SSxx = Ʃx² - (Ʃx)²/n = 198.5 - (34)²/6 =	5.833333333
SSyy = Ʃy² - (Ʃy)²/n = 0.5252 - (1.7)²/6 =	0.043533333
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 10.135 - (34)(1.7)/6 =	0.501666667

a)

Slope, b = SSxy/SSxx = 0.50167/5.83333 = 0.086

y-intercept, a = y̅ -b* x̅ = 0.28333 - (0.086)*5.66667 = -0.204

Regression equation :

ŷ = -0.204 + (0.086) x

b)

Null and alternative hypothesis:

Ho: ρ = 0

Ha: ρ > 0

α = 0.05

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 0.50167/√(5.83333*0.04353) = 0.9955

Test statistic :  

t = r*√(n-2)/√(1-r²) = 0.9955 *√(6 - 2)/√(1 - 0.9955²) = 21.0356

df = n-2 = 4

Critical value, t_c = T.INV(0.05, 4) = 2.1318

p-value = T.DIST.RT(21.0356, 4) = 0.0000

Conclusion:

p-value < α, Reject the null hypothesis. There is a positive correlation between pH levels and Optical Density at the 5% level of significance.