Statistic 10
Y can be most accurately predicted from X if the correlation between X and Y is _________.
- 0.98
Scatterplots are useful for several reasons.
- First, you can visually determine if the relationship is linear or non-linear. - Second, you can see if bivariate outliers exist. - Third, you can see the direction of the relationship.
When using correlation to describe a relationship, it is very important to remember that it only tells us the strength and direction of the relationship.
It does not tell us whether one variable caused or influenced the other.
That is correlation does not imply causation! Just because two things are related or associated does not necessarily mean once caused the other to change.
Of course, it also doesn't mean one variable didn't cause a change in the other. With correlational data, you just don't know.
There is a well documented relationship between saturated fat intake and heart disease. Does this mean saturated fat causes heart disease?
Probably not. Turns out saturated fat intake is also related to high levels of nitrates, trans-fats, sugar, etc. because most people get their saturated fat from hamburgers, chicken nuggets, Twinkies, and so forth. When people get their saturated fat from eggs, nuts, cream, and dairy, the relationship between saturated fat and heart disease largely disappears.
Typically, the explanatory or predictor variable is placed on the x-axis and the outcome or criterion variable is placed on the y-axis.
Remember, correlation does not mean causation, but you may suspect one variable is influencing the other. For example, there is a correlation between height and weight, but I doubt gaining weight makes you taller, so I would place height on the x-axis and weight on the y-axis.
both 1 and -1 represent perfect relationships. You will not find a perfect relationship when using psychological variables.
The relationship between a circle's diameter and circumference is perfect (positive). Two kids on a seesaw would represent a perfect negative relationship (as one goes up, the other must go down in the same amount).
Which of the following is (are) not correlation coefficients?
pearson r, eta, rho, phi
the pearson product moment correlation (r)
pearson's r allows us to quantify the relationship between two variables (must be interval or ratio) that are related in a linear fashion (a straight line best defines the relationship).
A relationship can be _________.
perfect, imperfect and nonexistant
A correlation of r = 0.60 exists between a set of X and Y scores. If a constant of 10 is added to each score of both distributions, the value of r will _______.
remain the same
first it tells us the strength of the relationship:
the further the number is away from 0, the stronger the relationship. For example, a .5 is stronger than a .4, while a -.8 is stronger than a .7 (the sign is irrelevant when indicating strength).
the regression line
the least-squares regression line is the unique line such that the sum of the squared vertical (y) distances between the data points and the line is the smallest possible
The coefficient of determination tells us how much variance in one variable that can be accounted for by a second.
true
The correlation between caloric intake and weight is positive.
true
The value of r obtained by calculating the correlation between X and Y is the same as the correlation between Y and X.
true
brivariate outliers may increase or decrease the value of r depending on the location
true
pearson r requires that the data be of interval or ratio scaling
true
the coefficient of determination can never be negative
true
the use of z scores allows comparisons between variables measured on different scales and units
true
by creating a scatter plot,
we are able to visually inspect the data to see if they are indeed related in a linear fashion. If they are not, the Pearson correlation method is not appropriate.
The line generated from this equation is called the line of best fit or least squares regression line,
which is the unique line that minimizes the distance from the line and the points on our graph.
If the correlation between two variables is -1.00 and the score of a given individual is 2.20 standard deviations above the mean on one of the variables, we would predict a score on the second variable of _________.
2.20 standard deviations below the mean
second, Pearson's r tells us in what direction the two variables are related by examining the sign in front of the number.
A negative tells us there is a negative(-) or inverse relationship, while a positive (+) tells us there is a positive relationship.
3rd variable problem
A problem that occurs when the researcher cannot directly manipulate variables; as a result, the researcher cannot be confident that another, unmeasured variable is not the actual cause of differences in the variables of interest.
coefficient of determination
a measure of the amount of variation in the dependent variable about its mean that is explained by the regression equation
In a positive relationship, _________.
a must be positive
After several studies, Professor Smith concludes that there is a zero correlation between body weight and bad tempers. This means that _________.
a person with a bad temper may be heavy or skinny
correlation coeeficient
a statistical measure that indicates the extent to which two factors vary together, and how well one factor can be predicted from the other - they may be positive or negative
Regression is
a statistical tool that allows us to predict one variable from another, although it originally meant regression towards mediocracy.
A scatterplot or scatter-graph is
a way to graph or display the correlation between two variables.
you have noticed that as people eat more ice cream they also have darker suntans. from this observation, you conclude _______.
all these are possible
Correlation and regression differ in that _________.
- correlation is primarily concerned with the size and direction of relationships - regression is primarily used for prediction
a traffic safety officer conducted an experiment to determine whether there is a correlation between people's ages and driving were randomly sampled and the following data were collected. age 20 25 45 46 60 65 speed 60 47 55 38 45 35the value of pearson r equals
-0.70
Which of the following values of r represents the strongest degree of relationship between two variables?
-0.80
The lowest degree of correlation shown below is _________.
0.15
If 49% of the total variability of Y is accounted for by X, what is the value of r?
0.70
Why?...because of something called the 3rd variable problem.
For example, the correlation between the number of churches in a town and the number of bars is r=.8, which is a strong, positive relationship. However, it is silly to assume more bars causes more churches or vice versa. A third variable, population, is responsible for the relationship. A larger town can support more bars and more churches, while a smaller town can support fewer bars and fewer churches.
In this equation, Y is the value of the variable you are trying to predict, and X is the value of the variable you know.
The symbol next to the X (whether it is a, m, or b) is always the slope of the line while the other constant is always the Y intercept.
the terms negative and positive in statistics do not mean bad and good.
The terms only refer to direction.
sometimes the Y will have a little "hat" on top (ŷ = 2.45 x -0.16).
This lets you know it is a predicted rather than an actual value.
Naturally, your prediction gets better the more closely related (correlated) the two variables are, which is why correlation and regression are often used together.
You can perfectly predict a circles diameter from its circumference because the two variable are perfectly related. Trying to predict someone's IQ from their shoe size would be useless since the two variables are not related.
pearson's r can assume values between -1 to +1,
and gives us two pieces of information (three if we square r to get the coefficient of determination)
The form of regression that is useful when using the Pearson's r is linear regression,
and it simply uses the equation for a straight line (remember, Pearson's r requires a linear relationship): Y=bX + a or Y=mX + b.
because two variables are used, it is referred to as a
bivariate statistic rather than a univariate (one variable).
Which of the following statements is true?
causation implies correlation
If there is no linear relationship between two variables, the value of r=-1.
false
Pearson's r requires a curvilinear relationship between variables.
false
The correlation coefficient must assume a value between 0 to +1.
false
instead of collecting data on the heights and weights of everyone in his school, joe only measured basketball players. as such, any correlation joe finds between height and weight is likely to be inflated.
false
An r of .4 is stronger than an r of .2, but how much stronger?
in order to answer this questions, you must first square r to get r2, which is called the coefficient of determination.
correlation
is a way to quantify the relationship (strength and direction) between two variables.
the relationship between speed and gas mileage, alcohol consumption and memory, and exercise and waist size are examples of
negative or inverse relationships.
the relationship between height and weight, study time and grades, and caloric intake and weight are
positively related.
The coefficient of determination
tells us how much variance in one variable can be accounted for by knowing another variable.
In a perfect positive correlation, each individual obtains the same z score on each variable.
true
A correlation between college entrance exam grades and scholastic achievement was found to be -1.08. On the basis of this you would tell the university that _________.
they should hire a new statistician
distances between the points and line are squared so all are positive values.
this is done so that distances can be properly added (pythagoras).
In a perfect linear relationship all the points must fall on a straight line.
true
