Lecture 14. Outline. Outline. Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA)

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1 Outline Lecture 14 Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA) 11-1 Introduction 11- Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination and Standard Error of Estimate Analysis of Variance (ANOVA) When an F test is used to test a hypothesis concerning the means of three or more populations, the technique is called analysis of variance (ANOVA) Assumptions for the F Test for Comparing Three or More Means The populations from which the samples were obtained must be normally or approximately normally distributed. The samples must be independent of each other. The variances of the populations must be equal Analysis of Variance Although means are being compared in this F test, variances are used in the test instead of the means. Two different estimates of the population variance are made.

2 1-4 4 Analysis of Variance Between-group variance - this involves computing the variance by using the means of the groups or between the groups. Within-group variance - this involves computing the variance by using all the data and is not affected by differences in the means Analysis of Variance The following hypotheses should be used when testing for the difference between three or more means. H 0 : µ 1 = µ = µ 3 = = µ k H 1 : At least one mean is different from the others Analysis of Variance Analysis of Variance -Example d.f.n. = k 1, where k is the number of groups. d.f.d. = N k, where N is the sum of the sample sizes of the groups. Note: The formulas for this test are tedious to work through, so examples will be done in MINITAB. See text for formulas. A marketing specialist wishes to see whether there is a difference in the average time a customer has to wait in a checkout line in three large self-service department stores. The times (in minutes) are shown on the next slide. Is there a significant difference in the mean waiting times of customers for each store using α = 0.05? Analysis of Variance -Example Analysis of Variance -Example Store A Store B Store C Step 1: State the hypotheses and identify the claim. H 0 : µ 1 = µ = µ 3 H 1 : At least one mean is different from the others (claim).

3 1-4 4 Analysis of Variance -Example Analysis of Variance -Example Step : Find the critical value. Since k = 3, N = 18, and α = 0.05, d.f.n. = k 1 = 3 1=, d.f.d. = N k = 18 3 = 15. The critical value is Step 3: Compute the test value. From the MINITAB output, F =.70. (See your text for computations). Step 4: Make a decision. Since.70 < 3.68, the decision is not to reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that there is a difference among the means. The ANOVA summary table is given on the next slide Analysis of Variance -Example Correlation and Regression 11-1 Introduction 11-1 Introduction Goal: to determine whether a relationship between two or more numerical or quantitative variables exists. Example: Is there a relationship between a person s age and his or her blood pressure? Correlation is a statistical method used to determine whether a 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.

4 11- Scatter plots The independent variable is the variable in regression that can be controlled or manipulated. The dependent variable is the variable in regression that cannot be controlled or manipulated. 11- Scatter Plots 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. 11- Scatter plots - example Is there a relationship between number of hours studied and test scores on an exam? 11- Scatter plots - example Independent variable: number of hours studied (x). Dependent variable: the grade the student received on the exam (y). The independent and dependent variables can be plotted on a graph called a scatter plot. 11- Scatter Plots - Example Construct a scatter plot for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data are shown in the following table. 11- Scatter Plots - Example Subject Age, x Pressure, y A B C D E F 70 15

5 11- Scatter Plots - Example 11- Scatter Plots - Other Examples Pressure Pressure Positive Relationship Age Age Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. The data are shown on the next slide. 11- Scatter Plots - Other Examples 11- Scatter Plots - Other Examples Final Final grade grade Negative Relationship Number Number of of absences absences Scatter Plots - Other Examples 11- Scatter Plots - Other Examples Construct a scatter plot for the data obtained in a study on the number of hours nine people exercise each week and the amount of milk (in ounces) each person consumes per week. The data follow.

6 Scatter Plots - Other Examples Correlation Coefficient y Y No Relationship x X The correlation coefficient computed from the sample data measures the strength and direction of a relationship between two variables. Sample correlation coefficient, r. Population correlation coefficient, ρ Range of Values for the Correlation Coefficient Formula for the Correlation Coefficient r Strong negative relationship No linear relationship Strong positive relationship r = n( xy) ( x)( y) ( ) ( ) [ n x x ][ n( y ) ( y) ] Where n is the number of data pairs Correlation Coefficient - Example (Verify) Compute the correlation coefficient for the age and blood pressure data. x = 345, y = 819, xy = 47, 634 x = 0, 399, y = 11, 443. Substituting in the formula for r gives r = The Significance of the Correlation Coefficient The population corelation coefficient, ρ, is the correlation between all possible pairs of data values (x, y) taken from a population.

7 The Significance of the Correlation Coefficient H 0 : ρ = 0 H 1 : ρ 0 This tests for a significant correlation between the variables in the population Formula for the t tests for the Correlation Coefficient n t=r 1 r with d. f. = n 11-3 Example Test the significance of the correlation coefficient for the age and blood pressure data. Use α = 0.05 and r = Step 1: State the hypotheses. H 0 : ρ = 0 H 1 : ρ Example Step : Find the critical values. Since α = 0.05 and there are 6 = 4 degrees of freedom, the critical values are t = and t =.776. Step 3: Compute the test value. t = (verify) Example Step 4: Make the decision. Reject the null hypothesis, since the test value falls in the critical region (4.059 >.776). Step 5: Summarize the results. There is a significant relationship between the variables of age and blood pressure. Correlation and Causation You must understand the nature of the linear relationship between the independent variable x and the dependent variable y. When a hypothesis test indicates that a significant linear relationship exists between the variables (i.e., when the null hypothesis has been rejected for a specific α value), any of the following five possibilities can exist.

8 Correlation and Causation #1 There is a direct cause-and-effect relationship between the variables. That is, x causes y. For example, water causes plants to grow, poison causes death, and heat causes ice to melt. Correlation and Causation # There is a reverse cause-and-effect relationship between the variables. That is, y causes x. Example: a researcher believes excessive coffee consumption causes nervousness, but the researcher fails to consider that the reverse situation may occur. That is, it may be that an extremely nervous person craves coffee to calm his or her nerves. Correlation and Causation #3 Correlation and Causation #4 The relationship between the variables may be caused by a third variable. For example, if a statistician correlated the number of deaths due to drowning and the number of cans of soft drink consumed during the summer, he or she would probably find a significant relationship. However, the soft drink is not necessarily responsible for the deaths, since both variables may be related to heat and humidity. There may be a complexity of interrelationships among many variables. For example, a researcher may find a significant relationship between students high school grades and college grades. But there probably are many other variables involved, such as IQ, hours of study, influence of parents, motivation, age, and instructors. Correlation and Causation #5 The relationship may be coincidental. For example, a researcher may be able to find a significant relationship between the increase in the number of people who are exercising and the increase in the number of people who are committing crimes. But common sense dictates that any relationship between these two values must be due to coincidence. Correlation and Causation Thus, when the null hypothesis is rejected, the researcher must consider all possibilities and select the appropriate one as determined by the study. Remember, correlation does not necessarily imply causation.

9 Regression The scatter plot for the age and blood pressure data displays a linear pattern. We can model this relationship with a straight line. This regression line is called the line of best fit or the regression line. The equation of the line is y = a + bx Regression Best fit means that the sum of the squares of the vertical distances from each point to the line is at a minimum. The reason one needs a line of best fit is that the values of y will be predicted from the values of x; hence, the closer the points are to the line, the better the fit and the prediction will be. See the next Figure. Regression the line of best fit Correlation coefficient and the line of best fit Formulas for the Regression Line y = a + bx. ( y)( x ) ( x)( xy) ( ) n x ) ( x) n( xy) ( x)( y) n( x ) ( x) a = b = Where a is the y intercept and b is the slope of the line Example Find the equation of the regression line for the age and the blood pressure data. Substituting into the formulas give a = and b = (verify). Hence, y = x. Note, a represents the intercept and b the slope of the line.

10 11-4 Example Using the Regression Line to Predict Pressure Pressure y = x Age 70 Age The regression line can be used to predict a value for the dependent variable (y) for a given value of the independent variable (x). Caution: Use x values within the experimental region when predicting y values Example Use the equation of the regression line to predict the blood pressure for a person who is 50 years old. Since y = x, then y = (50) = Note that the value of 50 is within the range of x values Coefficient of Determination and Standard Error of Estimate The coefficient of determination, denoted by r, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable Coefficient of Determination and Standard Error of Estimate Coefficient of Determination and Standard Error of Estimate r is the square of the correlation coefficient. The coefficient of nondetermination is (1 r ). Example: If r = 0.90, then r = The standard error of estimate, denoted by s est, is the standard deviation of the observed y values about the predicted y values. The formula is given on the next slide.

11 Formula for the Standard Error of Estimate Standard Error of Estimate - Example s or s est est = ( y y ) n y a y b xy = n From the regression equation, y = x and n = 6, find s est. Here, a = 55.57, b = 8.13, and n = 6. Substituting into the formula gives s est = 6.48 (verify) Prediction Interval A prediction interval is an interval constructed about a predicted y value, y, for a specified x value Prediction Interval For given α value, we can state with (1 α)100% confidence that the interval will contain the actual mean of the y values that correspond to the given value of x Formula for the Prediction Interval about a Value y Prediction interval - Example y t s α est y + t s α est 1 n( x X ) 1+ + n n x x < y < with d. f. = n ( ) 1 n( x X ) 1+ + n n x x ( ) A researcher collects the data shown on the next slide and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y = x. Find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old.

12 Prediction Interval - Example Machine Age, x (Years) Monthly cost, y A 1 $6 B $78 C 3 $70 D 4 $90 E 4 $93 F 6 $ Prediction Interval - Example Step 1: Find Σx, Σx and X. Σx = 0, Σx 0 = 8, X = = Step : Find y for x = 3. y = (3) = Step 3: Find s est s est = 6.48 as shown in previous example Prediction Interval - Example Biomathematics Step 4: Substitute in the formula and solve. t α/ =.776, d.f. = 6 = 4 for 95% < y < (verify) Hence, one can be 95% confident that the interval < y < contains the actual value of y. János FODOR That Is All 00