A medical researcher wanted to determine the effect of pH (a measure of alkalinity or acidity, with pure water having a pH of 7) on the growth of a bacteria culture.
The table below gives the measurements of different cultures, in thousands of bacteria, after 8 hours.
pH Number of bacteria
(in thousands)
4 112
5 117
6 128
7 134
8 142
9 151
10 146
11 162
(a) Find the linear regression equation for these data. (Round your coefficients to three decimal places.
y = _________
(b) Using the regression model, what is the expected number of bacteria when the pH is 6.5? Round to the nearest thousand bacteria.
______ bacteria
To find the linear regression equation for the given data, we need to determine the equation in the form of y = mx + b, where y is the dependent variable (number of bacteria), x is the independent variable (pH), m is the slope, and b is the y-intercept.
Step 1: Calculate the means of x and y:
Mean of x (pH):
Sum of pH values / total number of pH values
Mean of y (number of bacteria):
Sum of bacteria values / total number of bacteria values
In this case:
Sum of pH values = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 60
Sum of bacteria values = 112 + 117 + 128 + 134 + 142 + 151 + 146 + 162 = 1092
Mean of x = 60 / 8 = 7.5
Mean of y = 1092 / 8 = 136.5
Step 2: Calculate the deviations of x and y:
Deviation of x (pH):
Subtract the mean of x from each pH value
Deviation of y (number of bacteria):
Subtract the mean of y from each bacteria value
In this case:
Deviation of x:
4 - 7.5 = -3.5
5 - 7.5 = -2.5
6 - 7.5 = -1.5
7 - 7.5 = -0.5
8 - 7.5 = 0.5
9 - 7.5 = 1.5
10 - 7.5 = 2.5
11 - 7.5 = 3.5
Deviation of y:
112 - 136.5 = -24.5
117 - 136.5 = -19.5
128 - 136.5 = -8.5
134 - 136.5 = -2.5
142 - 136.5 = 5.5
151 - 136.5 = 14.5
146 - 136.5 = 9.5
162 - 136.5 = 25.5
Step 3: Calculate the sum of the products of deviations:
Sum of the products of deviations = Σ((deviation of x) * (deviation of y))
(-3.5 * -24.5) + (-2.5 * -19.5) + (-1.5 * -8.5) + (-0.5 * -2.5) + (0.5 * 5.5) + (1.5 * 14.5) + (2.5 * 9.5) + (3.5 * 25.5) = 336.25
Step 4: Calculate the sum of the squared deviations of x:
Sum of the squared deviations of x = Σ((deviation of x)^2)
(-3.5)^2 + (-2.5)^2 + (-1.5)^2 + (-0.5)^2 + (0.5)^2 + (1.5)^2 + (2.5)^2 + (3.5)^2 = 56
Step 5: Calculate the slope (m) of the regression equation:
m = (sum of the products of deviations) / (sum of the squared deviations of x)
m = 336.25 / 56 = 6.008
Step 6: Calculate the y-intercept (b) of the regression equation:
b = (mean of y) - (slope * mean of x)
b = 136.5 - (6.008 * 7.5) = 89.85
Therefore, the linear regression equation for these data is:
y = 6.008x + 89.85
(a) The linear regression equation is y = 6.008x + 89.85.
(b) To find the expected number of bacteria when the pH is 6.5, substitute x = 6.5 into the linear regression equation:
y = 6.008 * 6.5 + 89.85
y = 39.052 + 89.85
y = 128.902
Rounding to the nearest thousand bacteria, the expected number of bacteria when the pH is 6.5 is 129,000 bacteria.