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Quiz Unit 3

Find the proportion of observations from a standard Normal distribution that fall in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region.

1. z<0.83

2. z>-1.0

3. z>-1.56

Examining Relationships

When are some situations when we might want to examine a relationship between two variables?

- Height & Heart Attacks
- Weight & Blood Pressure
- Hours studying & test scores
- What else?

In this chapter we will deal with relationships and quantitative variables; the next chapter will deal with more categorical variables.

The response variable is our dependent variable (traditionally y)

- The explanatory variable is our independent variable (traditionally x)

Which is the explanatory and which is the response variable?

Jim wants to know how the mean 2005 SAT Math and Verbal scores in the 50 states are related to each other. He doesn\'t think that either score explains or causes the other.

Julie looks at some data. She asks, “Can I predict a state\'s mean 2005 SAT Math score if I know its mean 2005 SAT Verbal score?”

Explanatory or Response?When we deal with cause and effect, there is always a definite response variable and explanatory variable.

But calling one variable response and one variable explanatory doesn\'t necessarily mean that one causes change in the other.

Explanatory and Response VariablesWhen analyzing several-variable data, the same principles appy…Data Analysis Toolbox

To answer a statistical question of interest involving one or more data sets, proceed as follows.

- DATA

Organize and examine the data. Answer the key questions.

- GRAPHS

Construct appropriate graphical displays.

- NUMERICAL SUMMARIES

Calculate relevant summary statistics

- INTERPRETATION

Look for overall patterns and deviations

When the overall pattern is regular, use a mathematical model to describe it.

Let\'s say we wanted to examine the relationship between the percent of a state\'s high school seniors who took the SAT exam in 2005 and the mean SAT Math score in state that year. A scatterplot is an effective way to graphically represent our data.

But first, what is the explanatory variable and what is the response variable in this situation?

ScatterplotsOnce we decide on the response and explanatory variables, we can create a scatterplot.Scatterplots

response variable

explanatory variable

Scatterplot Tips

- Plot the explanatory variable on the horizontal axis. If there is no explanatory-response distinctions, either variable can go on the horizontal axis.
- Label both axes!
- Scale the horizontal and vertical axes. The intervals must be uniform.
- If you are given a grid, try to adopt a scale so that your plot uses the whole grid. Make your plot large enough so that the details can be easily seen.

negative

Interpreting Scatterplots

- Direction?
- Form?
- Strength?
- Outliers?

The Mean SAT Math scores and percent of hish school seniors who take the test, by state, with the southern states highlighted.

Is the South different?

Adding Categorical DataLinear relations are important because, when we discuss the relationship between two quantitative variables, a straight line is a simple pattern that is quite common.

A strong linear relationship has points that lie close to a straight line.

A weak linear relationship has points that are widely scattered about a line.

Measuring Linear Association:CorrelationOur eyes are not good measures of how strong a linear relationship is, so...

A numerical measure along with a graph gives the linear association an exact value.

In words, standardize each value, multiply corresponding values, add them up, and divide by n-1

Facts about Correlation

- Correlation makes no distinction between explanatory and response variables.
- r doesn\'t change when we change the units of measurement of x, y, or both.
- r is positive when the association is positive and is negative when the association is negative.
- The correlation r is always a number between -1 and 1. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 toward either -1 or 1.

Cautionary Notes about Correlation

- Correlation requires that both variables be quantitative.

- Correlation does not describe curved relationships, no matter how strong they are.

- Like the mean and standard deviation, the correlation is not resistant; r is strongly affected by a few outlying observations.

- Correlation is not a complete summary of two-variable data. You should give the means and standard deviations of both x and y along with the correlation.

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