Fiveable
Fiveable

or

Log in

Find what you need to study


Light

Find what you need to study

Unit 9 Overview: Slopes

7 min readjanuary 8, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

"[You] may be surprised to learn that there is in . In [your probable] experience in previous courses, the of the does not vary for a particular set of . However, suppose that every student in a university physics course collects data on spring length for 10 different hanging masses and calculates the for their sample data. The students’ slopes would likely vary as part of an approximately normal centered at the (true) of the relating spring length to hanging mass.

In this unit, [you'll] learn how to construct for and perform about the of a when appropriate conditions are met!" -- College Board, AP Statistics Course Description

Recap time!

In Unit 3, you got introduced to concepts related to : , y-intercept, R^2, standard deviation of the residual s, and in context from computer output. We also placed huge emphasis on avoiding (e.g., “a 1-foot increase in X is associated with a 0.445-point increase in Y”) in favor of framing the association in terms of (i.e., “a predicted 0.445-point increase”). Now, we'll apply what we've learned in the past couple units and connect slopes to inferential statistics! 😌

Recap Time: What is "Inference"?

Remember from Units 6-8, that is a huuuuge part of statistics. In fact, it is the most important and useful part of the AP Statistics course (and it's also tested very heavily). is the act of using a sample to either make a or test a claim about a . 🔮

In Unit 8, we looked at a more complicated way of doing for by using procedures for with multiple categories (data presented in a two way table). In this unit, we are going to look at a more complex procedure for quantitative data by looking at bivariate data instead of univariate data. Therefore, our data will be presented in a .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fstats9-mml6XCXLpDbv.png?alt=media&token=e57f6478-4dae-4e24-839e-9c576c411978

Source: Chartio

Scatterplots

A is the most useful way to display . One variable for each data point is displayed on the x-axis and one variable is displayed on the y-axis. As seen in the example above, this often allows us to see correlation or patterns in our data points. 📈

Explanatory Variable

The explanatory variable, or , is the variable that is typically found along the x axis. One way to remember this is that this variable is what does the “explaining” of the patterns we are generally seeing in the overall layout of the .

Response Variable

The , or , is the variable that is typically found along the y axis. One way to remember this is that this variable “responds” to the other variable in building our pattern. Also, it “depends” on the other variable.

Example Variable

For example, let’s say we are investigating the correlation between shoe size and height. Ask yourself, “Does shoe size depend on height, or does height depend on shoe size?” While either of these make sense and would give a similar pattern on a , it is more reasonable to say that someone’s shoe size typically depends on their height. Therefore, shoe size is the and should be found on the y axis.

Inference with Scatterplots

Again, as you can recall from Unit 2, our models have several parts: a , y-intercept, r value, and R^2 value. While an r value and R^2 value do a good job at determining how correlated our points are along a , they don’t quite give us the procedure with hypotheses and being able to say that there is evidence of correlation. 🕵️

This is where our and t-test for a come in to give us not just one value for a , but a range of possible values that we can be confident contains the true of our regression model rather than just one .

T-Interval

As with all of the other units involving , the first form of procedure we are doing is constructing a confidence interval. A confidence interval is a form of that allows us to predict the true of our regression line. While our sample gives us one , adding in one point can change this model greatly. So rather than honing in on just one , adding a to that gives us a range of values that we can be pretty certain contains the true of our model for all possible points. 🤺

T-Test

The other form of in scatterplots is a t-test. In this type of , we will be testing a null hypothesis that states that the between our two variables is 0 (ie, there is no relation). After running our test, we can determine if there is enough evidence that hypothesis needs rejecting in favor of another hypothesis (that the is not 0). This type of test closely correlates with our r value in that the stronger our r value, the more likely we are to reject our null hypothesis. 📝

Big Questions in This Unit

Some questions that you'll be able to answer at the end of unit 9. Here are some possible answers we came up and see if they makes sense (or you'd like to contest them) later on!

  • 💡 How can there be in if the statistic is uniquely determined for a ?

The of a is not uniquely determined, but rather it is estimated based on sample data. Therefore, it is possible for the estimated to vary from sample to sample. This is why it is appropriate to perform about the of a based on sample data, because the sample is only an estimate of the true population .

  • 💡 When is it appropriate to perform about the of a based on sample data?

It's appropriate to perform about the of a based on sample data whenever you want to draw conclusions about the of the based on your sample data. This is typically done when you want to test whether there is a significant linear relationship between two variables in the population, or when you want to estimate the strength and direction of this relationship.

For example, you might collect data on the heights and weights of a sample of people, and then use this data to estimate the of the that describes the relationship between height and weight in the population. If you find that the of the sample regression line is significantly different from zero, you can conclude that there is a significant linear relationship between height and weight in the population.

Note that in order to perform statistical about the of a , you must assume that your sample is representative of the population and that the sample data follows a certain statistical model (e.g., the model).

  • 💡 Why do we not conclude that there is no correlation between two variables based on the results of a statistical for slopes?

We don't conclude that there is no correlation between two variables based on the results of a statistical for slopes because the of a only represents the strength and direction of the linear relationship between the two variables. It is possible for there to be a nonlinear relationship between two variables, or for there to be no relationship at all, even if the of the is not significantly different from zero.

For example, consider the case where there is a strong quadratic relationship between two variables, but the of the is not significantly different from zero. In this case, the of the would not be a good indicator of the strength and direction of the relationship between the two variables, and you would not be able to conclude that there is no correlation between the two variables based on the results of a statistical for slopes!

Circling back to the big picture of this unit, it's important to consider the nature of the relationship between two variables and to use appropriate statistical methods to test for and quantify this relationship, rather than relying solely on the of a .

🎥  Watch: AP Stats Unit 9 - Inference for Slopes

Key Terms to Review (25)

Bivariate Quantitative Data

: Bivariate quantitative data refers to a set of data that consists of two quantitative variables. These variables are measured or observed for each individual or case in the dataset.

Categorical Data

: Categorical data refers to data that can be divided into categories or groups based on qualitative characteristics.

Confidence Intervals

: Confidence intervals are ranges of values calculated from sample data that are likely to contain an unknown population parameter with a certain level of confidence.

Dependent Variable

: The dependent variable is the outcome or result that depends on changes made to other variables, particularly on changes made intentionally by researchers (independent variables).

Deterministic Language

: Deterministic language refers to the use of precise and unambiguous terms in statistical explanations. It avoids vague or subjective wording, ensuring that statements are clear and can be replicated by others.

Independent Variable

: The independent variable is the variable that is intentionally manipulated or changed by the researcher in an experiment. It is the cause or input in a study.

Inference

: Inference involves drawing conclusions or making predictions about a population based on sample data. It allows us to make generalizations and statements about a larger group using information from a smaller subset.

Least-Squares Regression Line

: The least-squares regression line is a straight line that best fits the pattern of bivariate quantitative data by minimizing the sum of squared differences between the observed values and predicted values based on the line.

Line of Best Fit

: The line of best fit is a straight line that represents the overall trend or relationship between two variables in a scatter plot. It is used to make predictions and estimate values.

Linear Regression

: Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. It helps us understand how changes in one variable are associated with changes in another variable.

Margin of Error

: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.

Population Parameter

: A population parameter is a numerical value that describes a characteristic of an entire population.

Population Regression Line

: The population regression line is a straight line that represents the average relationship between an independent variable and a dependent variable in a population.

Potential Outcomes

: Potential outcomes refer to the different outcomes that could occur under different treatment conditions in a study. It involves comparing what happens to individuals when they receive different treatments or interventions.

Prediction

: Prediction involves using available information to estimate or forecast future events or outcomes. It uses patterns and relationships in data to make educated guesses about what might happen next.

Response Variable

: The response variable is the outcome or result that researchers measure and analyze in an experiment. It represents the effect or output of interest.

Sampling Distribution

: A sampling distribution refers to the distribution of a statistic (such as mean, proportion, or difference) calculated from multiple random samples taken from the same population. It provides information about how sample statistics vary from sample to sample.

Scatterplot

: A scatterplot is a graph that displays the relationship between two quantitative variables. It uses dots to represent individual data points and shows how they are distributed along the x and y axes.

Significance Tests

: Significance tests help determine whether an observed effect or difference between groups is statistically significant or simply due to chance variation.

Slope

: Slope represents how steep or flat a line is. In statistics, it specifically refers to how much one variable changes for every unit change in another variable.

Standard Deviation of the Residual (s)

: The standard deviation of the residual, denoted as s, measures the average distance between each observed data point and the regression line. It tells us how much the actual data points deviate from the predicted values.

Standard Error of the Slope

: The standard error of the slope measures how much we expect different samples to vary in terms of their slopes when fitting regression lines. It quantifies how accurate our estimate for the slope is based on sample data.

t-interval for slopes

: The t-interval for slopes is a statistical method used to estimate the range of possible values for the slope of a regression line. It provides a confidence interval that helps determine if there is a significant relationship between two variables.

Two-Way Table

: A two-way table organizes categorical data for two variables and shows how they are related to each other.

Variability

: Variability refers to the spread or dispersion of data points in a dataset. It measures how much the values differ from each other.

Unit 9 Overview: Slopes

7 min readjanuary 8, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

"[You] may be surprised to learn that there is in . In [your probable] experience in previous courses, the of the does not vary for a particular set of . However, suppose that every student in a university physics course collects data on spring length for 10 different hanging masses and calculates the for their sample data. The students’ slopes would likely vary as part of an approximately normal centered at the (true) of the relating spring length to hanging mass.

In this unit, [you'll] learn how to construct for and perform about the of a when appropriate conditions are met!" -- College Board, AP Statistics Course Description

Recap time!

In Unit 3, you got introduced to concepts related to : , y-intercept, R^2, standard deviation of the residual s, and in context from computer output. We also placed huge emphasis on avoiding (e.g., “a 1-foot increase in X is associated with a 0.445-point increase in Y”) in favor of framing the association in terms of (i.e., “a predicted 0.445-point increase”). Now, we'll apply what we've learned in the past couple units and connect slopes to inferential statistics! 😌

Recap Time: What is "Inference"?

Remember from Units 6-8, that is a huuuuge part of statistics. In fact, it is the most important and useful part of the AP Statistics course (and it's also tested very heavily). is the act of using a sample to either make a or test a claim about a . 🔮

In Unit 8, we looked at a more complicated way of doing for by using procedures for with multiple categories (data presented in a two way table). In this unit, we are going to look at a more complex procedure for quantitative data by looking at bivariate data instead of univariate data. Therefore, our data will be presented in a .

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fstats9-mml6XCXLpDbv.png?alt=media&token=e57f6478-4dae-4e24-839e-9c576c411978

Source: Chartio

Scatterplots

A is the most useful way to display . One variable for each data point is displayed on the x-axis and one variable is displayed on the y-axis. As seen in the example above, this often allows us to see correlation or patterns in our data points. 📈

Explanatory Variable

The explanatory variable, or , is the variable that is typically found along the x axis. One way to remember this is that this variable is what does the “explaining” of the patterns we are generally seeing in the overall layout of the .

Response Variable

The , or , is the variable that is typically found along the y axis. One way to remember this is that this variable “responds” to the other variable in building our pattern. Also, it “depends” on the other variable.

Example Variable

For example, let’s say we are investigating the correlation between shoe size and height. Ask yourself, “Does shoe size depend on height, or does height depend on shoe size?” While either of these make sense and would give a similar pattern on a , it is more reasonable to say that someone’s shoe size typically depends on their height. Therefore, shoe size is the and should be found on the y axis.

Inference with Scatterplots

Again, as you can recall from Unit 2, our models have several parts: a , y-intercept, r value, and R^2 value. While an r value and R^2 value do a good job at determining how correlated our points are along a , they don’t quite give us the procedure with hypotheses and being able to say that there is evidence of correlation. 🕵️

This is where our and t-test for a come in to give us not just one value for a , but a range of possible values that we can be confident contains the true of our regression model rather than just one .

T-Interval

As with all of the other units involving , the first form of procedure we are doing is constructing a confidence interval. A confidence interval is a form of that allows us to predict the true of our regression line. While our sample gives us one , adding in one point can change this model greatly. So rather than honing in on just one , adding a to that gives us a range of values that we can be pretty certain contains the true of our model for all possible points. 🤺

T-Test

The other form of in scatterplots is a t-test. In this type of , we will be testing a null hypothesis that states that the between our two variables is 0 (ie, there is no relation). After running our test, we can determine if there is enough evidence that hypothesis needs rejecting in favor of another hypothesis (that the is not 0). This type of test closely correlates with our r value in that the stronger our r value, the more likely we are to reject our null hypothesis. 📝

Big Questions in This Unit

Some questions that you'll be able to answer at the end of unit 9. Here are some possible answers we came up and see if they makes sense (or you'd like to contest them) later on!

  • 💡 How can there be in if the statistic is uniquely determined for a ?

The of a is not uniquely determined, but rather it is estimated based on sample data. Therefore, it is possible for the estimated to vary from sample to sample. This is why it is appropriate to perform about the of a based on sample data, because the sample is only an estimate of the true population .

  • 💡 When is it appropriate to perform about the of a based on sample data?

It's appropriate to perform about the of a based on sample data whenever you want to draw conclusions about the of the based on your sample data. This is typically done when you want to test whether there is a significant linear relationship between two variables in the population, or when you want to estimate the strength and direction of this relationship.

For example, you might collect data on the heights and weights of a sample of people, and then use this data to estimate the of the that describes the relationship between height and weight in the population. If you find that the of the sample regression line is significantly different from zero, you can conclude that there is a significant linear relationship between height and weight in the population.

Note that in order to perform statistical about the of a , you must assume that your sample is representative of the population and that the sample data follows a certain statistical model (e.g., the model).

  • 💡 Why do we not conclude that there is no correlation between two variables based on the results of a statistical for slopes?

We don't conclude that there is no correlation between two variables based on the results of a statistical for slopes because the of a only represents the strength and direction of the linear relationship between the two variables. It is possible for there to be a nonlinear relationship between two variables, or for there to be no relationship at all, even if the of the is not significantly different from zero.

For example, consider the case where there is a strong quadratic relationship between two variables, but the of the is not significantly different from zero. In this case, the of the would not be a good indicator of the strength and direction of the relationship between the two variables, and you would not be able to conclude that there is no correlation between the two variables based on the results of a statistical for slopes!

Circling back to the big picture of this unit, it's important to consider the nature of the relationship between two variables and to use appropriate statistical methods to test for and quantify this relationship, rather than relying solely on the of a .

🎥  Watch: AP Stats Unit 9 - Inference for Slopes

Key Terms to Review (25)

Bivariate Quantitative Data

: Bivariate quantitative data refers to a set of data that consists of two quantitative variables. These variables are measured or observed for each individual or case in the dataset.

Categorical Data

: Categorical data refers to data that can be divided into categories or groups based on qualitative characteristics.

Confidence Intervals

: Confidence intervals are ranges of values calculated from sample data that are likely to contain an unknown population parameter with a certain level of confidence.

Dependent Variable

: The dependent variable is the outcome or result that depends on changes made to other variables, particularly on changes made intentionally by researchers (independent variables).

Deterministic Language

: Deterministic language refers to the use of precise and unambiguous terms in statistical explanations. It avoids vague or subjective wording, ensuring that statements are clear and can be replicated by others.

Independent Variable

: The independent variable is the variable that is intentionally manipulated or changed by the researcher in an experiment. It is the cause or input in a study.

Inference

: Inference involves drawing conclusions or making predictions about a population based on sample data. It allows us to make generalizations and statements about a larger group using information from a smaller subset.

Least-Squares Regression Line

: The least-squares regression line is a straight line that best fits the pattern of bivariate quantitative data by minimizing the sum of squared differences between the observed values and predicted values based on the line.

Line of Best Fit

: The line of best fit is a straight line that represents the overall trend or relationship between two variables in a scatter plot. It is used to make predictions and estimate values.

Linear Regression

: Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. It helps us understand how changes in one variable are associated with changes in another variable.

Margin of Error

: The margin of error is a measure of the uncertainty or variability in survey results. It represents the range within which the true population parameter is likely to fall.

Population Parameter

: A population parameter is a numerical value that describes a characteristic of an entire population.

Population Regression Line

: The population regression line is a straight line that represents the average relationship between an independent variable and a dependent variable in a population.

Potential Outcomes

: Potential outcomes refer to the different outcomes that could occur under different treatment conditions in a study. It involves comparing what happens to individuals when they receive different treatments or interventions.

Prediction

: Prediction involves using available information to estimate or forecast future events or outcomes. It uses patterns and relationships in data to make educated guesses about what might happen next.

Response Variable

: The response variable is the outcome or result that researchers measure and analyze in an experiment. It represents the effect or output of interest.

Sampling Distribution

: A sampling distribution refers to the distribution of a statistic (such as mean, proportion, or difference) calculated from multiple random samples taken from the same population. It provides information about how sample statistics vary from sample to sample.

Scatterplot

: A scatterplot is a graph that displays the relationship between two quantitative variables. It uses dots to represent individual data points and shows how they are distributed along the x and y axes.

Significance Tests

: Significance tests help determine whether an observed effect or difference between groups is statistically significant or simply due to chance variation.

Slope

: Slope represents how steep or flat a line is. In statistics, it specifically refers to how much one variable changes for every unit change in another variable.

Standard Deviation of the Residual (s)

: The standard deviation of the residual, denoted as s, measures the average distance between each observed data point and the regression line. It tells us how much the actual data points deviate from the predicted values.

Standard Error of the Slope

: The standard error of the slope measures how much we expect different samples to vary in terms of their slopes when fitting regression lines. It quantifies how accurate our estimate for the slope is based on sample data.

t-interval for slopes

: The t-interval for slopes is a statistical method used to estimate the range of possible values for the slope of a regression line. It provides a confidence interval that helps determine if there is a significant relationship between two variables.

Two-Way Table

: A two-way table organizes categorical data for two variables and shows how they are related to each other.

Variability

: Variability refers to the spread or dispersion of data points in a dataset. It measures how much the values differ from each other.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.