Session 21 Notes, Section 10-1

Scatter Plots and Correlation

In this semester we have learned about descriptive statistics and inferential statistics.

And in the last half of the semester, we leaned about two areas of inferential statistics: hypothesis testing and confidence intervals.

The last area of inferential statistics we are studying this semester is determining whether a relationship exists between two or more numerical or quantitative variables.

Many relationships among variables exist in the real world. One way to determine whether a relationship exists is to use the statistical techniques known as correlation and regression.  In this session we will explore graphing and study correlation to determine if a relationship exists.

At the end of this session you will be able to:

• Understand the terms and vocabulary of correlation and regression
• Draw a scatter plot for a set of ordered pairs.
  • Example
• Compute the correlation coefficient.
  • Example
• Test the hypothesis H₀: p= 0.
  • Example
  • Example
  • Example
• Your Turn:  Summary

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Vocabulary

• Statistical Methods
  • Correlation is a statistical method used to determine whether a linear relationship between variables exists.
  • Regression is a statistical method used to describe the nature of the relationship between variables—that is, positive or negative, linear or nonlinear.
• Statistical Questions:  These are questions that we ask in inferential statistics:
  • Are two or more variables related?
  • If so, what is the strength of the relationship?
  • What type or relationship exists?
  • What kind of predictions can be made from the relationship?
• Vocabulary
  • A correlation coefficient is a measure of how variables are related.
  • In a simple relationship, there are only two types of variables under study.
  • In multiple relationships, many variables are under study.

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Draw a scatter plot for a set of ordered pairs

When you finish this lesson you will be able to

• Draw a scatter plot on graph paper
• Draw a scatter plot using your calculator.

Draw scatter plot on graph paper.

• A scatter plot is a graph of the ordered pairs (x,y) of numbers consisting of the independent variable, x, and the dependent variable, y.
• A scatter plot is a visual way to describe the nature of the relationship between the independent and dependent variables.  In the following scatter plot, we have a relationship between hours of exercises for the independent variable and ounces of milk for a dependent variable.
• 

Draw a scatter plot using your graphing calculator.

Go to the end of Section 10-2 in your textbook. With your calculator in hand, go through the steps, one step at a time, for creating a scatter plot as shown on page 562 - 563.  Use Example TI10-1.  (Do not go beyond preparing a scatter  plot.)

1.  Enter the x values in L₁ and the y values in L₂.

• Press "Stat".  The calculator will display EDIT, CALC, AND TESTS across the top of the screen.
• Select 1: Edit.  The top of the screen should display L1, L2, L3 etc.
• Now enter your values for x under L1 and values for y under L2.

2.  Set the Window size using the instructions from the bottom of page 562.  (The Xscl and Yscl determine how close together the "tick" marks are on the horizontal and vertical scales.)"

3.  The "STAT PLOT F₁" key is with the "Y=" key.  After you press 2nd [STAT PLOT F₁] for Plot 1, press the ENTER key.

4.  Using the right arrow key, move the curser to On and press enter.

5.  Step 5 is done as in the textbook. ("Mark" presents two ways the plotted points can be displayed.)

6.  Step 6 is done as in the textbook. (""GRAPH" is the right key directly under the screen.)

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Example of Scatter Plot

Create the scatter plot for the following set of data associated with a study of age and systolic blood pressure of six people selected at random:

The scatter plot would look like the graph below.  With your algebraic knowledge of plotting points, identify each point corresponding the an ordered pair from the table above.

Notice the relationship is linear, like a straight line, and that the pressure goes up as age goes up.

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Pearson Product Moment Correlation Coefficient

When you finish this lesson you will be able to

• Describe the characteristics of the Pearson Correlation Coefficient
• Compute Pearson Correlation Coefficient using the formula
• Compute Pearson Correlation Coefficient using your calculator
• Rounding Pearson Correlation Coefficient
• Interpreting the Pearson Correlation Coefficient

Characteristics of the Pearson Correlation Coefficient

• The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables.
  • The symbol for the sample correlation coefficient is r.
  • The symbol for the population correlation coefficient is the Greek letter ρ (rho).
• The range of the correlation coefficient is from 1 to 1.
  • If there is a strong positive linear relationship between the variables, the value of r or ρ will be close to 1.
  • If there is a strong negative linear relationship between the variables, the value of r or ρ will be close to-1.
  • When there is no linear relationship between the variables or only a weak relationship, the value of r or ρ will be close to 0.
  • 

Formula for the Correlation Coefficient r

There are several formulas for the correlation coefficient.  We will use the most commonly used formula called the Pearson Product Moment Correlation Coefficient:

• 
  • where n is the number of data pairs

Calculate the Correlation Coefficient r with your calculator

Go to page 520 of Section 10-4 in your textbook. With your calculator in hand, go through the steps, one step at a time, as outlined below .  Use Example TI10-1 at the top of page 521.  (Do not go beyond finding the correlation coefficient.)

1.  Enter the x values in L₁ and the y values in L₂.

• Press "Stat".  The calculator will display EDIT, CALC, AND TESTS across the top of the screen.
• Select 1: Edit.  The top of the screen should display L1, L2, L3 etc.
• Now enter your values for x under L1 and values for y under L2.

2.  Make sure your Diagnostic Display Mode is on.

• Press "2nd" "Catalog".  ("Catalog" is under the "0" key at the bottom of your calculator.)
• Use the down arrow key to scroll down the menu until the little arrow points to "DiagnosticsOn".
• Press "Enter".  You should see "DiagnosticsOn" displayed on the screen.
• Press "Enter" again.

Once you have done this, you will not have to do it again unless the calculator is reset to factory specifications. 

3.  Obtain the correlation coefficient.

• Press "Stat".
• Use right arrow key to highlight CALC.
• Press "8" or scroll down to "8" and press "ENTER".
• Press "2nd LIST" and press ENTER for L₁.(Or select a different list if your data is someplace else.)
• Press ",".
• Press "2nd LIST", press "2", and press ENTER for L₂. (Or select a different list if your data is someplace else.)

Now your correlation coefficient r will be the last number displayed on your screen.

Rounding the Correlation Coefficient r

Round the correlation  coefficient r to three decimal places so that it can be compared to critical values in Table L.  Make sure that only the final answer is rounded, not intermediate calculations.

Interpreting the Significance of Correlation Coefficient r

Interpretations of being close to 0 or 1 is vague.  Thus we use this specific criterion:  If the absolute value of  the computed value of  r exceeds the value in Table L, we conclude there is a significant correlation coefficient.  Otherwise, there is not sufficient evidence to support the conclusion of a significant correlation coefficient.

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Example of Correlation Coefficient

Compute the value of the correlation coefficient for the data obtained in the study of age and blood pressure.

• 
• Create a table to calculate the various elements of the formula

  • Subject	Age x	Blood Pressure y	xy	x^2	y^2
    A	44	125	5500	1936	15625
    B	41	122	5002	1681	14884
    C	56	135	7560	3136	18225
    D	61	144	8784	3721	20736
    E	68	145	9860	4624	21025
    F	77	151	11627	5929	22801
    6	347	822	48333	21027	113296

• Calculate r

  • Numerator of r	square of denominator	r
    4764	23541276	0.981876

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Test the hypothesis H₀: p= 0.

• Formally defined, the population correlation coefficient, ρ, is the correlation computed by using all possible pairs of data values (x, y) taken from a population
  • The sample correlation coefficient can be used as an estimator of he population correlation coefficient if the following assumptions are valid:
    • The variables x and y are linearly related
    • The variables are random variables
    • The two variables have a bivariate normal distribution
      • (given any value of x, the y variable is normally distribted)
• Traditional (Classical) Hypothesis-testing procedure
  1. State the hypotheses
  2. Find the critical values
  3. Compute the test value
  4. Make the decision
  5. Summarize the results
• In hypothesis testing, one of the following is true:
  • H₀: ρ = 0
    • This null hypothesis means that there is no correlation between the x and y variables in the population.
  • H₁: ρ not = 0
    • This alternative hypothesis means that there is a significant correlation between the variables in the population.

• Formula for the t test for the correlation coefficient:

  • 
    • with degrees of freedom equal to n - 2.
  • Example

• P-value method

  1. State the hypotheses.
  2. Find the test value.
  3. Find the P-value
  4. Make the decision
  5. Summarize the results
     • Example

• Use the PPMC (Pearson product moment correlation coefficient) table for the correlation coefficient that are significant for a specific alpha level and a specific number of degrees of freedom.  NOTE: Table I ( p 738) in our book is only for two tailed alpha for .05 or .01  and is only for selected d.f. past 20.  (However you will like to use this table compared to the other approaches for hypothesis testing of H₀!)

  1. State the hypotheses.
  2. Find the critical value.
  3. Make the decision
  4. Summarize the results

     • Example

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Example.

Test the significance of the correlation coefficient for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data are shown in the table.

Use α= 0.20, and r = 0.949.

1. State the hypotheses.
   • H₀: ρ = 0  and H₁: ρ not = 0
2. Find the critical values.
   • Since α= 0.20 and there are 6 - 2 = 4 degrees of freedom, the critical values obtained from the t Distribution table (two tailed on page 731) are 1.533 and -1.533
3. Compute the test value.
   • 
4. Make the decision.
   • Reject the null hypothesis, since the test value falls in the critical region.
5. Summarize the results.
   • There is a significant relationship between the variables of age and blood pressure.

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Example.

Test the significance of the correlation coefficient for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data are shown in the table.

Use α= 0.01, and r = 0.889.

1. State the hypotheses.
   • H₀: ρ = 0  and H₁: ρ not = 0
2. Find the test value.
   • 
   • Since there are 6 - 2 = 4 degrees of freedom, look for 3.883 on the t Distribution table (two tailed on page 731 in the d.f. = 4 row).  3.883 is greater than 3.747 and less than 4.604.
3. Find the P-value
   • 0.01 < P-value < .02 (read the α values from the two tailed row above 3.747 and 4.604)
4. Make the decision.
   • Since  0.01 < 0.01 < .02 is not a true statement, reject the null hypothesis.  (NOTE: if our test value had been 3.747 instead of 3.883, the P-value would have been equal to 0.01 and there would not have been a range and we would have not rejected the null hypothesis)
5. Summarize the results.
   • There is a significant relationship between the variables of age and blood pressure.

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Example.

The following data were obtained in a study on the number of hours that nine people exercise each week and the amount of milk (in ounces) each person consumes each week.

1. State the hypotheses.
   • H₀: ρ = 0  and H₁: ρ not = 0
   •  
2. Find the Critical Value
   • Since the sample size is 9, there are 7 degrees of freedom.
   • When = 0.01 and with 7 degrees of freedom, the value obtained from table I (page 738)  for the critical values for PPMC is 0.798.
   • For a significant relationship, a value of r greater than + 0.798 or less than - 0.798 is needed.
3. Decision
   •  Since r = 0.294 and 0.294 des not fall in the critical ranges for a significant relationship, the null hypothesis is not rejected.
   • 
4. Summary
   • There is not enough evidence to say that there is a significant linear relationship between the variables.

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Summary

• The strength and direction of the linear relationship between variables is measured by the value of the correlation coefficient r.
• r can assume values between and including 1 and -1.
• The closer the value of the correlation coefficient is to 1 or -1, the stronger the linear relationship is between the variables.
• A value of 1 or -1 indicates a perfect linear relationship.

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