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)
4 115
5 115
6 132
7 141
8 141
9 151
10 147
11 168
(a) Find the linear regression equation for these data. (Round your coefficients to three decimal places. A graphing calculator is recommended.)
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 can use a graphing calculator or statistical software. Here's how you can do it using a graphing calculator:
Step 1: Enter the data points into the calculator. Let the pH values be the x-values and the number of bacteria be the y-values.
Step 2: Go to the regression feature on your calculator. Typically, you can find it in the STAT (Statistics) menu.
Step 3: Choose linear regression (often denoted as LinReg or LinReg(ax+b)).
Step 4: Select the list of x-values (pH) and the list of y-values (number of bacteria).
Step 5: Calculate the regression equation.
After performing these steps, the calculator will give you the equation in the form of y = ax + b, where a and b are coefficients.
(a) The linear regression equation for the given data will take the form "y = mx + b" or "y = ax + b". The equation can be written as follows:
y = 2.404x + 112.405
(Rounded to three decimal places)
(b) To find the expected number of bacteria when the pH is 6.5, substitute the pH value into the regression equation and solve for y:
y = 2.404(6.5) + 112.405
Calculating this equation will give you the expected number of bacteria when the pH is 6.5.
The expected number of bacteria when the pH is 6.5 is approximately 126.