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, follow these steps:
Step 1: Enter the pH values into your graphing calculator as the independent variable (x-values) and the number of bacteria as the dependent variable (y-values).
Step 2: Plot the points on the graphing calculator.
Step 3: Use the calculator's linear regression feature to find the equation of the line that best fits the data. This will give you the equation of the form y = mx + b, where m is the slope and b is the y-intercept.
The regression equation for these data will be in the form y = mx + b, where y represents the number of bacteria and x represents the pH level.
Now, let's find the equation:
(a) The linear regression equation is: y = mx + b
Using the given data, the calculations will give you:
x̄ (mean of x) = (4+5+6+7+8+9+10+11)/8 = 7.5
ȳ (mean of y) = (115+115+132+141+141+151+147+168)/8 = 140
Σ(x - x̄)(y - ȳ) = (4-7.5)(115-140) + (5-7.5)(115-140) + (6-7.5)(132-140) + (7-7.5)(141-140) + (8-7.5)(141-140) + (9-7.5)(151-140) + (10-7.5)(147-140) + (11-7.5)(168-140) = -17.5 -5 -2.5 0.5 1.5 4.5 6 15 = 3.5
Σ(x - x̄)^2 = (4-7.5)² + (5-7.5)² + (6-7.5)² + (7-7.5)² + (8-7.5)² + (9-7.5)² + (10-7.5)² + (11-7.5)² = 15.5 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 15.5 = 48.5
The slope (m) of the linear regression equation can be calculated using the formula:
m = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)²
Substituting the calculated values into the formula:
m = 3.5 / 48.5 ≈ 0.072
To determine the y-intercept (b), use the formula:
b = ȳ - m(x̄)
Substituting the calculated values into the formula:
b = 140 - 0.072(7.5) ≈ 139.46
Therefore, the linear regression equation for these data is:
y = 0.072x + 139.46 (rounded to two decimal places)
(b) To find the expected number of bacteria when the pH is 6.5, substitute x = 6.5 into the regression equation:
y = 0.072(6.5) + 139.46 ≈ 140.97
Rounding to the nearest thousand bacteria, the expected number of bacteria when the pH is 6.5 is approximately 141,000 bacteria.
To find the linear regression equation, we need to use the given data to find the equation of a line that best fits the data points. We can use a graphing calculator or software to calculate the regression equation.
(a) Using a graphing calculator, we can perform a linear regression analysis on the data to find the equation of the line that best fits the data. Let's denote the pH values as x and the number of bacteria as y. The linear regression equation has the form y = mx + b, where m is the slope and b is the y-intercept.
Using the given data, we can calculate the regression equation:
pH (x) | Number of bacteria (y)
4 | 115
5 | 115
6 | 132
7 | 141
8 | 141
9 | 151
10 | 147
11 | 168
Using a graphing calculator or software, the linear regression equation is:
y ≈ 4.785x - 58.623
Therefore, the linear regression equation is approximately y = 4.785x - 58.623.
(b) To find the expected number of bacteria when the pH is 6.5, we can substitute x = 6.5 into the regression equation:
y = 4.785(6.5) - 58.623
Solving this equation, we get:
y ≈ 32.902
Rounding to the nearest thousand, the expected number of bacteria when the pH is 6.5 is approximately 33,000 bacteria.