Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W.
Math Chapter 4 ● weak associations result in a large amount of scatter in the scatterplot ● the stronger the association, the better the model is for prediction ● variation is not important when studying scatterplots ● correlation coefficients can range from -1 to 1 ○ its not possible to have a correlation coefficient outside this range ● the correlation coefficient is a measure that describes the direction and strength of a linear relationship ○ can be used for two quantitative variables only ● a correlation coefficient of zero indicates that there is no linear relationship between the two quantitative variables ● the mean of the data set is often the best estimate of the parameter ○ the best estimate is also the best prediction ● regression is used to find the equation that best fits the data ○ it finds the equation that minimizes the error in the dependent variable for the given values of the predictor variable ● a regression line gives a linear equation relating the independent and dependent variable ○ allows predictions of the dependent variable to be made based on the value of the independent variable ○ gives us a better prediction than just the mean ● linear relationships are not curved ● the correlation coefficient remains the same when the numbers are multiplied by a positive constant ● the correlation coefficient remains the same when a constant is added to each number ● positive correlation means that both variables tend to increase or decrease together ● negative correlation means that two variables tend to change in opposite directions, with one increasing while the other decreases ● no correlation means that there is no apparent relationship between the two variables ● how we determine the strength of a correlation: the more closely two variables follow the general trend, the stronger the correlation, which may be positive or negative ● the correlation coefficient is a number that measure the strength of the linear association between two numerical variables ○ represented by r ○ always a number between -1 and 1 ● the correlation coefficient makes sense only if the trend is linear and the variables are numerical ● a correlation coefficient based on observational study can never be used to support a claim of cause and effect ● when computing a correlation coefficient, changing the order of the variables does not change r ○ in an equation, it does not matter which variable is called x and which is called y ● if there is no trend, the value of r is near zero ● the regression equation is a tool for making predictions about future observed values. It also provides a useful way of summarizing a linear relationship ● statisticians often write the word “predicted” in front of the y-variable in the equation of the regression line to emphasize that the line consists of predictions for the y-variable, not actual values ● another name for the regression line is the least squares line ● regression lines make predictions about the values of y for a given x-value ● an influential point is a point that changes the regression equation by a large amount ● correlation does not imply causation ○ just because two variables are related does not show that one caused the other ● extrapolation means that the regression line is used to make predictions beyond the range of the data ● regression towards the mean = if a variable is extreme on its first measurement, it will tend to be closer to the average on a second measurement ○ if it is extreme on the second, it will tend to have been closer to the average on the first ● regression models are useful only for linear associations ● outliers = influential points ● extrapolation = attempting to use the regression equation to make predictions beyond the range of data ● coefficient of determination = the value that measure how much variation in the response variable is explained by the explanatory variable ○ r squared = correlation coefficient squared
