Data points that are outliers in the horizontal direction are more likely to be influential than. If the point in the upper right corner of this scatterplot is removed from the data set, then what will happen to the slope b of the line of best fit and to the correlation r?

Two variables that are actually not related to each other may nonetheless have a very high correlation because they both result from some other, possibly hidden, factor. This is an example of. The figure below is a scatterplot of reading test scores against IQ test scores for 14 fifth-grade children. There is one low outlier in the plot. What effect does this low outlier have on the correlation?

In a statistics course, a linear regression equation was computed to predict the final exam score based on the score on the first test of the term.

George scored 80 on the first test. On the final exam George scored What is the value of his residual? The equation of the least squares regression line for a set of points in a scatterplot is given above. The point 5, 7 is one point on this scatterplot. Which of the following is the residual for the point 5, 7? For the yearsthe number of heart disease deaths perpeople in the United States were recorded.

Calamba spa with extra serviceThe regression line below was computed using a statistical software package. The regression line estimates that for every 3. For every 3.

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The number of heart disease deaths perpeople has been dropping by an estimated 3. Data obtained from a group of high school seniors comparing age and the number of hours spent on the telephone.

The resulting regression equation is. What percentage of the variation in the number of hours spent on the telephone can be explained by this least-squares regression model? A researcher wishes to examine the relationship between years of schooling completed and the number of pregnancies in young women.

Her research discovers a linear relationship, and a least squares fit for her data results in the model above, where x is the number of years completed in school and y is the number of pregnancies.

Diablo 2 weaponsWhat is the estimated change in the number of pregnancies that corresponds to the completion of an additional 10 years of school? Played 53 times. Print Share Edit Delete. Live Game Live. Finish Editing. This quiz is incomplete! To play this quiz, please finish editing it. Delete Quiz. Question 1. Which of the following statements is not true of the correlation r between the lengths in inches and weights in pounds of a sample of brook trout?

Write the linear regression equation for the data set foot length x and height y. In order to linearize data that is growing exponentially, you must take:.

What's the appropriate interpretation of the slope of the regression line? For every additional 0.Linear regression, also known as simple linear regression or bivariate linear regression, is used when we want to predict the value of a dependent variable based on the value of an independent variable. The dependent variable can also be referred to as the outcome, target or criterion variable, whilst the independent variable can also be referred to as the predictor, explanatory or regressor variable.

We will refer to these as dependent and independent variables throughout this guide. For example, you could use linear regression to understand whether test anxiety can be predicted based on revision time i. Alternatively, you could use linear regression to understand whether cholesterol concentration a fat in the blood linked to heart disease can be predicted based on time spent exercising i. Note: If you have two or more independent variables, rather than just one, you need to use multiple regression.

Alternatively, if you just want to establish whether a linear relationship exists, but are not making predictions, you could use Pearson's correlation. If your dependent variable is dichotomous, you could use a binomial logistic regression.

In this guide, we show you how to carry out linear regression using Minitab, as well as interpret and report the results from this test. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for linear regression to give you a valid result. We discuss these assumptions next.

Linear regression has seven assumptions. You cannot test the first two of these assumptions with Minitab because they relate to your study design and choice of variables. However, you should check whether your study meets these assumptions before moving on. If these assumptions are not met, there is likely to be a different statistical test that you can use instead.

Assumptions 1 and 2 are explained below:.

**Regression Analysis (Testing Significance Of Independent Variables,T-Stat, P-Value, Etc.)**

Assumptions 3, 4, 5, 6 and 7 relate to the nature of your data and can be checked using Minitab. You have to check that your data meets these assumptions because if it does not, the results you get when running a linear regression might not be valid. In fact, do not be surprised if your data violates one or more of these assumptions. This is not uncommon. However, there are possible solutions to correct such violations e. Assumptions 3, 4, 5, 6 and 7 are explained below:.

In practice, checking for assumptions 3, 4, 5, 6 and 7 will probably take up most of your time when carrying out linear regression. However, it is not a difficult task, and Minitab provides all the tools you need to do this. In the section, Test Procedure in Minitabwe illustrate the Minitab procedure required to perform linear regression assuming that no assumptions have been violated.

First, we set out the example we use to explain the linear regression procedure in Minitab. An educator wants to determine whether students' exam scores were related to revision time.

For example, as students spent more time revising, did their exam score also increase a positive relationship ; or did the opposite happen? The educator also wanted to know the proportion of exam score that revision time could explain, as well as being able to predict the exam score. The educator could then determine whether, for example, students that spent just 10 hours revising could still pass their exam. Therefore, the dependent variable was "exam score", measured on a scale from 0 toand the independent variable was "revision time", measured in hours.

To carry out the analysis, the researcher recruited 40 students. The length of time revising i. Expressed in variable terms, the researcher wanted to regress Exam score on Revision time. A linear regression was used to determine whether there was a statistically significant relationship between exam score and revision time.Rogers Sponsor.

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To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Course summary. Analyzing categorical data. Welcome to AP Statistics : Analyzing categorical data Analyzing one categorical variable : Analyzing categorical data Two-way tables : Analyzing categorical data. Distributions in two-way tables : Analyzing categorical data Mosaic plots : Analyzing categorical data.

Displaying and describing quantitative data. Frequency tables and dot plots : Displaying and describing quantitative data Histograms and stem-and-leaf plots : Displaying and describing quantitative data Describing and comparing distributions : Displaying and describing quantitative data. Summarizing quantitative data. Measuring center in quantitative data : Summarizing quantitative data More on mean and median : Summarizing quantitative data Measuring spread in quantitative data : Summarizing quantitative data.

More on standard deviation optional : Summarizing quantitative data Box and whisker plots : Summarizing quantitative data. Modeling data distributions. Percentiles cumulative relative frequency : Modeling data distributions Z-scores : Modeling data distributions Effects of linear transformations : Modeling data distributions. Density curves : Modeling data distributions Normal distributions and the empirical rule : Modeling data distributions Normal distribution calculations : Modeling data distributions.

Exploring bivariate numerical data. Making and describing scatterplots : Exploring bivariate numerical data Correlation coefficients : Exploring bivariate numerical data Least-squares regression equations : Exploring bivariate numerical data.

### AP®︎/College Statistics

Assessing the fit in least-squares regression : Exploring bivariate numerical data. Study design. Sampling and observational studies : Study design Sampling methods : Study design Types of studies experimental vs. Experiments : Study design. Randomness, probability, and simulation : Probability Addition rule : Probability Multiplication rule : Probability.

Conditional probability : Probability. Random variables. Discrete random variables : Random variables Continuous random variables : Random variables Transforming random variables : Random variables. Combining random variables : Random variables Binomial random variables : Random variables Binomial mean and standard deviation formulas : Random variables Geometric random variables : Random variables. Sampling distributions. What is a sampling distribution? Sampling distribution of a sample mean : Sampling distributions Sampling distributions for differences in sample means : Sampling distributions.

Confidence intervals. Introduction to confidence intervals : Confidence intervals Confidence intervals for proportions : Confidence intervals Confidence intervals for means : Confidence intervals. Significance tests hypothesis testing. The idea of significance tests : Significance tests hypothesis testing Error probabilities and power : Significance tests hypothesis testing Testing hypotheses about a proportion : Significance tests hypothesis testing.

Testing hypotheses about a mean : Significance tests hypothesis testing. Inference comparing two groups or populations. Confidence intervals for the difference between two proportions : Inference comparing two groups or populations Testing the difference between two proportions : Inference comparing two groups or populations Confidence intervals for the difference between two means : Inference comparing two groups or populations. Testing the difference between two means : Inference comparing two groups or populations.

Chi-square tests for categorical data. Chi-square goodness-of-fit tests : Chi-square tests for categorical data Chi-square tests for relationships : Chi-square tests for categorical data. More on regression.It is unlikely that you will be asked to draw a scatterplot on the exam; however, being familiar with creating and analyzing a scatterplot of bivariate data could prove crucial. Make sure that your regression diagnostics are turned on.

Be aware that, when fitting a linear regression model to a set of data, the residuals are automatically stored in a list called RESID. Remember to investigate a residual plot to confirm whether your proposed regression models the data well.

Refer to the video below for steps for doing this on your calculator. The computation of probabilities for random variables that are normally distributed is a core concept in the course. You can expect a question involving a normal probability computation on the AP Statistics exam. The normalcdf command takes inputs of the mean, standard deviation, and an lower and upper boundary. This is helpful because it means you do not need to standardize turn to z-scores the values before using the command.

To find the probability that X takes a value between 65 and 86 if X is normally distributed with mean 70 and standard deviation 15, follow the steps in the video below. You know the verbal cues to recognize when a question is about a binomial context, but which probability menu item do you use for a given situation? If the question is asking for the probability of a single outcome i.

If the question is asking about the probability of more than 5 heads in 10 coin flips, then you need binomcdf. Keep in mind that binomcdf is cumulative from 0 to k successes, so pay close attention to whether to subtract the endpoint, or not, when using the compliment rule. Always remember that for the free-response section you must show the correct work and justification for the computation.

This video shows an example of this kind of computation. In some cases, you are asked to calculate the mean or standard deviation of a random variable given in a table. You can use a little-known feature of the 1-Var Stat command to generate the summary statistics for a random variable by putting the values of the variable in one list and the probabilities in another list.

In the example below, L1 was the values of the variable and L2 was the list of probabilities. Rarely on the exam will you be asked directly to make a graph of univariate data. But, when checking the conditions for an inferential procedure involving means, knowing quickly how to generate a graph of data and determine its relative shape is important. Use a modified boxplot option 4 in the StatPlots menu or a histogram option 3 to assess the shape of a sample distribution for strong deviations from approximately normal.

Use ZoomStat to set the window quickly. More steps are included in the video below. As sure as the sun will rise tomorrow, you will be asked to carry out an inference procedure on the free response portion of the exam. You may be given sample statistics to work with or perhaps a small data set to input. Either way, you must be sure to know which type of inferential procedure to do. Be sure to be familiar with each type of inferential procedure, including its outputs.

Recall that the t-procedures are required when dealing with questions about a population mean or difference in two population means.Jamison's Website. Search this site. Helpful Websites Math Dictionary. Geometry Reference. Free Algebra Worksheets.

Free Geometry Worksheets. Free Algebra 2 Worksheets. Courses Honors Precalculus. AP Statistics. Algebra 1. Honors Algebra 2. Statistics Mrs.

Carrie Jamison cjamison johnbapst. Students who enroll in this course are required to take the AP Statistics exam in May.

The four major areas covered by the course are exploring data, planning a study, anticipating patterns, and statistical inference. Course Syllabus. Formulas and Tables.

Link to Digital Copy of Textbook. Mad Libs for Statistics. Practice Multiple Choice Exam. Preparing for the Exam:. Video : Formula Sheet Review. Website : Stuff not on the Formula Sheet. Statistics Project Description. Part 1: Exploring and Understanding Data. Video : Comparing Dot Plots, histograms, and Boxplots.

Video : Chapter 5 Comparing Distributions Sarah. Video : Finding Outliers 1. Video : Checking for Outliers. Video : How to Find Outliers. Part 2: Exploring Relationships Between Variables.

What does it all Mean? Part 3: Gathering Data. Video : Convenience Sampling. Video : Caution about Sample Surveys. Video : Statistical Significance of Experiment. Video : Randomized Block Design for Experiments. Part 4: Randomness and Probability. Video : At least One Probabilities. Video : Coin Flipping Probability, Including "at least one".In a cause and effect relationship, the independent variable is the cause, and the dependent variable is the effect. Least squares linear regression is a method for predicting the value of a dependent variable Ybased on the value of an independent variable X.

If you view this web page on a different browser e. For the next few lessons, we focus on the case where there is only one independent variable. This is called simple regression. Toward the end of the tutorial, we will cover multiple regression, which handles two or more independent variables.

Tip: The next lesson presents a simple linear regression example that shows how to apply the material covered in this lesson. Since this lesson is a little dense, you may benefit by also reading the next lesson. Linear regression finds the straight line, called the least squares regression line or LSRL, that best represents observations in a bivariate data set.

Suppose Y is a dependent variable, and X is an independent variable. The population regression line is:. Normally, you will use a computational tool - a software package e.

Agitator selection criteriaYou enter the X and Y values into your program or calculator, and the tool solves for each parameter. In the unlikely event that you find yourself on a desert island without a computer or a graphing calculator, you can solve for b 0 and b 1 "by hand". Here are the equations.

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When the regression parameters b 0 and b 1 are defined as described above, the regression line has the following properties. The least squares regression line is the only straight line that has all of these properties. The coefficient of determination denoted by R 2 is a key output of regression analysis.

It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable. The formula for computing the coefficient of determination for a linear regression model with one independent variable is given below.

Coefficient of determination. The coefficient of determination R 2 for a linear regression model with one independent variable is:. The standard error about the regression line often denoted by SE is a measure of the average amount that the regression equation over- or under-predicts. The higher the coefficient of determination, the lower the standard error; and the more accurate predictions are likely to be.

A researcher uses a regression equation to predict home heating bills dollar costbased on home size square feet.

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