Science: Interpreting Scatter Plots and Linear Regressions
Using data trends, best-fit lines, slopes, and residuals in science investigations
Science: Interpreting Scatter Plots and Linear Regressions
Using data trends, best-fit lines, slopes, and residuals in science investigations
Science - Grade 9-12
- 1
A biology class records plant height after different numbers of days under a grow light. The scatter plot shows that plant height generally increases as days increase. Describe the relationship between the variables and state whether the association is positive, negative, or no association.
Look for whether the points rise, fall, or stay scattered as you move from left to right.
The relationship is positive because plant height tends to increase as the number of days under the grow light increases. This means the variables move in the same direction. - 2
A linear regression model for plant growth is h = 1.8d + 4.2, where h is plant height in centimeters and d is time in days. What does the slope 1.8 mean in this science context?
The slope means the plant height is predicted to increase by about 1.8 centimeters for each additional day under the grow light. - 3
For the regression model T = -0.65a + 24, T is water temperature in degrees Celsius and a is altitude in hundreds of meters. Interpret the slope in context.
A negative slope means the predicted y-value decreases as the x-value increases.
The slope means the water temperature is predicted to decrease by 0.65 degrees Celsius for every increase of 100 meters in altitude. - 4
A chemistry group measures reaction rate at different temperatures. Their regression equation is r = 0.12T - 1.5, where r is reaction rate in moles per second and T is temperature in degrees Celsius. Predict the reaction rate at 30 degrees Celsius.
Substituting 30 for T gives r = 0.12(30) - 1.5 = 3.6 - 1.5 = 2.1. The predicted reaction rate is 2.1 moles per second. - 5
A scatter plot compares hours of sunlight per day with daily solar panel energy output. The points are closely clustered around an upward-sloping line. What can you conclude about the strength and direction of the relationship?
Strong relationships have points close to a clear pattern.
The relationship is strong and positive because the points are close to an upward-sloping pattern. More sunlight hours are associated with higher solar panel energy output. - 6
A student studies the relationship between shoe size and score on a biology test. The scatter plot has points spread randomly with no clear pattern. Is a linear regression model useful for prediction in this case? Explain.
A linear regression model would not be very useful because there is no clear linear relationship between shoe size and biology test score. Predictions from the model would likely be unreliable. - 7
A regression model for dissolved oxygen in a stream is O = -0.35T + 14.1, where O is dissolved oxygen in milligrams per liter and T is water temperature in degrees Celsius. Predict the dissolved oxygen when the water temperature is 20 degrees Celsius.
Multiply the slope by the x-value, then add the y-intercept.
Substituting 20 for T gives O = -0.35(20) + 14.1 = -7 + 14.1 = 7.1. The predicted dissolved oxygen is 7.1 milligrams per liter. - 8
A scatter plot shows the relationship between the mass of a pendulum bob and the period of the pendulum. The points form a nearly horizontal cloud. What does this suggest about mass and period in this experiment?
This suggests that the mass of the pendulum bob has little or no linear relationship with the period in this experiment. Changing the mass does not appear to strongly change the period. - 9
A regression model predicts bacterial colony count as C = 52t + 120, where C is the number of colonies and t is time in hours. What is the y-intercept, and what does it mean in context?
The y-intercept is the predicted value when x equals 0.
The y-intercept is 120. It means the model predicts there were 120 bacterial colonies at time 0 hours. - 10
A physics lab collects the following data for spring stretch: force in newtons: 1, 2, 3, 4, 5 and stretch in centimeters: 2.1, 4.0, 6.2, 7.9, 10.1. Describe the trend and estimate a reasonable linear regression equation.
The data show a strong positive linear trend because stretch increases as force increases. A reasonable regression equation is approximately stretch = 2.0(force) + 0.1, meaning the spring stretches about 2.0 centimeters per newton. - 11
A student uses the model pH = -0.04c + 7.2, where c is carbon dioxide concentration in parts per million. The class measured concentrations only from 300 ppm to 600 ppm. Is it appropriate to use this model to predict pH at 2000 ppm? Explain.
Models are usually most reliable within the range of data used to create them.
It is not appropriate because 2000 ppm is far outside the measured data range. Using a regression model far beyond the data is extrapolation and may give an unreliable prediction. - 12
In an ecology study, a regression line predicts 18 bird species for a forest patch of 12 hectares. The actual observed number is 15 species. Calculate the residual and interpret it.
The residual is actual minus predicted, so 15 - 18 = -3. The residual is -3 species, meaning the model overpredicted the number of bird species by 3. - 13
A scatter plot of air temperature versus cricket chirps per minute has a regression line with equation C = 4.5T - 80, where C is chirps per minute and T is temperature in degrees Celsius. The line fits the points fairly well. What temperature does the model predict when crickets chirp 55 times per minute?
This time the y-value is given, so solve the equation for the x-value.
Set 55 = 4.5T - 80. Add 80 to both sides to get 135 = 4.5T. Divide by 4.5 to get T = 30. The predicted temperature is 30 degrees Celsius. - 14
A student reports that the correlation between fertilizer amount and plant biomass is r = 0.92. What does this value indicate about the relationship?
Correlation values close to 1 show strong positive linear relationships.
A correlation of 0.92 indicates a strong positive linear relationship. Higher fertilizer amounts are associated with higher plant biomass in the data. - 15
A scatter plot compares distance from a factory with lead concentration in soil. The regression line slopes downward, and one point close to the factory has much higher lead than the other points. Explain how this outlier could affect the regression line and the interpretation.
The outlier could pull the regression line upward near small distances and make the relationship appear stronger or steeper than it is for most of the data. Scientists should investigate the outlier and consider whether it represents a real measurement or an error.