The F Distribution and the F-Ratio The distribution used for the hypothesis test is a new one. It is called the F distribution , invented by George Snedecor but named in honor of Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator. For example, if F follows an F distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F 4,10 . To calculate the F ratio , two estimates of the variance are made. Variance between samples : An estimate of σ 2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation. Variance within samples : An estimate of σ 2 that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation. SS between = the sum of squares that represents the variation among the different samples SS within = the sum of squares that represents the variation within samples that is due to chance. To find a "sum of squares" means to add together squared quantities that, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics . MS means " mean square ." MS between is the variance between groups, and MS within is the variance within groups. Calculation of Sum of Squares and Mean Square k = the number of different groups n j = the size of the j th group s j = the sum of the values in the j th group n = total number of all the values combined (total sample size: ∑ n j ) x = one value: ∑ x = ∑ s j Sum of squares of all values from every group combined: ∑ x 2 Total sum of squares: S S total = ∑ x 2 ( ∑ x ) 2 n Explained variation: sum of squares representing variation among the different samples: SS between = ∑ [ ( s j ) 2 n j ] − ( ∑ s j ) 2 n Unexplained variation: sum of squares representing variation within samples due to chance: S S within = S S total – S S between df 's for different groups ( df 's for the numerator): d f between = k – 1 Equation for errors within samples ( df 's for the denominator): df within = n – k Mean square (variance estimate) explained by the different groups: MS between = S S between d f between Mean square (variance estimate) that is due to chance (unexplained): MS within = S S within d f within MS between and MS within can be written as follows: M S between = S S between d f between = S S between k − 1 M S w i t h i n = S S w i t h i n d f w i t h i n = S S w i t h i n n − k The one-way ANOVA test depends on the fact that MS between can be influenced by population differences among means of the several groups. Since MS within compares values of each group to its own group mean, the fact that group means might be different does not affect MS within . The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MS between and MS within should both estimate the same value. NOTE The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution, because it is assumed that the populations are normal and that they have equal variances. F -Ratio or F Statistic F = M S between M S within If MS between and MS within estimate the same value (following the belief that H 0 is true), then the F -ratio should be approximately equal to one. Mostly, just sampling errors would contribute to variations away from one. As it turns out, MS between consists of the population variance plus a variance produced from the differences between the samples. MS within is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MS between will generally be larger than MS within .Then the F -ratio will be larger than one. However, if the population effect is small, it is not unlikely that MS within will be larger in a given sample. The foregoing calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F -ratio can be written as: F -Ratio Formula when the groups are the same size F = n ⋅ s x – 2 s 2 pooled where ... n = the sample size df numerator = k – 1 df denominator = n – k s 2 pooled = the mean of the sample variances (pooled variance) s x – 2 = the variance of the sample means Data are typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software. Source of variation Sum of squares ( SS ) Degrees of freedom ( df ) Mean square ( MS ) F Factor (Between) SS (Factor) k – 1 MS (Factor) = SS (Factor)/( k – 1) F = MS (Factor)/ MS (Error) Error (Within) SS (Error) n – k MS (Error) = SS (Error)/( n – k ) Total SS (Total) n – 1 Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The one-way ANOVA results are shown in . Plan 1: n 1 = 4 Plan 2: n 2 = 3 Plan 3: n 3 = 3 5 3.5 8 4.5 7 4 4 3.5 3 4.5 s 1 = 16.5, s 2 =15, s 3 = 15.5 Following are the calculations needed to fill in the one-way ANOVA table. The table is used to conduct a hypothesis test. S S ( b e t w e e n ) = ∑ [ ( s j ) 2 n j ] − ( ∑ s j ) 2 n = s 1 2 4 + s 2 2 3 + s 3 2 3 − ( s 1 + s 2 + s 3 ) 2 10 where n 1 = 4, n 2 = 3, n 3 = 3 and n = n 1 + n 2 + n 3 = 10 = ( 16.5 ) 2 4 + ( 15 ) 2 3 + ( 15.5 ) 2 3 − ( 16.5 + 15 + 15.5 ) 2 10 S S ( b e t w e e n ) = 2.2458 S ( t o t a l ) = ∑ x 2 − ( ∑ x ) 2 n = ( 5 2 + 4.5 2 + 4 2 + 3 2 + 3.5 2 + 7 2 + 4.5 2 + 8 2 + 4 2 + 3.5 2 ) − ( 5 + 4.5 + 4 + 3 + 3.5 + 7 + 4.5 + 8 + 4 + 3.5 ) 2 10 = 244 − 47 2 10 = 244 − 220.9 S S ( t o t a l ) = 23.1 S S ( w i t h i n ) = S S ( t o t a l ) − S S ( b e t w e e n ) = 23.1 − 2.2458 S S ( w i t h i n ) = 20.8542 Source of variation Sum of squares ( SS ) Degrees of freedom ( df ) Mean square ( MS ) F Factor (Between) SS (Factor) = SS (Between) = 2.2458 k – 1 = 3 groups – 1 = 2 MS (Factor) = SS (Factor)/( k – 1) = 2.2458/2 = 1.1229 F = MS (Factor)/ MS (Error) = 1.1229/2.9792 = 0.3769 Error (Within) SS (Error) = SS (Within) = 20.8542 n – k = 10 total data – 3 groups = 7 MS (Error) = SS (Error)/( n – k ) = 20.8542/7 = 2.9792 Total SS (Total) = 2.2458 + 20.8542 = 23.1 n – 1 = 10 total data – 1 = 9 Try It As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments bare soil a commercial ground cover black plastic straw compost All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the n = 15 plants: Bare: n 1 = 3 Ground Cover: n 2 = 3 Plastic: n 3 = 3 Straw: n 4 = 3 Compost: n 5 = 3 2,625 5,348 6,583 7,285 6,277 2,997 5,682 8,560 6,897 7,818 4,915 5,482 3,830 9,230 8,677 Create the one-way ANOVA table. Enter the data into lists L1, L2, L3, L4 and L5. Press STAT and arrow over to TESTS. Arrow down to ANOVA. Press ENTER and enter L1, L2, L3, L4, L5). Press ENTER. The table was filled in with the results from the calculator. One-Way ANOVA table: Source of variation Sum of squares ( SS ) Degrees of freedom ( df ) Mean square ( MS ) F Factor (Between) 36,648,561 5 – 1 = 4 36 , 648 , 561 4 = 9 , 162 , 140 9 , 162 , 140 2 , 044 , 672.6 = 4.4810 Error (Within) 20,446,726 15 – 5 = 10 20 , 446 , 726 10 = 2 , 044 , 672.6 Total 57,095,287 15 – 1 = 14 The one-way ANOVA hypothesis test is always right-tailed because larger F -values are way out in the right tail of the F -distribution curve and tend to make us reject H 0 . Let’s return to the slicing tomato exercise in . The means of the tomato yields under the five mulching conditions are represented by μ 1 , μ 2 , μ 3 , μ 4 , μ 5 . We will conduct a hypothesis test to determine if all means are the same or at least one is different. Using a significance level of 5%, test the null hypothesis that there is no difference in mean yields among the five groups against the alternative hypothesis that at least one mean is different from the rest. The null and alternative hypotheses are: H 0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 H a : μ i ≠ μ j some i ≠ j The one-way ANOVA results are shown in Source of variation Sum of squares ( SS ) Degrees of freedom ( df ) Mean square ( MS ) F Factor (Between) 36,648,561 5 – 1 = 4 36,648,561 4 = 9,162,140 9,162,140 2,044,672 .6 = 4 .4810 Error (Within) 20,446,726 15 – 5 = 10 20,446,726 10 = 2,044,672 .6 Total 57,095,287 15 – 1 = 14 Distribution for the test: F 4,10 df ( num ) = 5 – 1 = 4 df ( denom ) = 15 – 5 = 10 Test statistic: F = 4.4810 Probability Statement: p -value = P ( F > 4.481) = 0.0248. Compare α and the p -value: α = 0.05, p -value = 0.0248 Make a decision: Since α > p -value, we cannot accept H 0 . Conclusion: At the 5% significance level, we have reasonably strong evidence that differences in mean yields for slicing tomato plants grown under different mulching conditions are unlikely to be due to chance alone. We may conclude that at least some of mulches led to different mean yields. Try It There are multiple variants of the virus that causes COVID-19. The length of hospital stays for patients afflicted with various strains of COVID-19 is shown in . Delta Strain Omicron Strain Alpha Strain Gamma Strain Beta Strain 13.9 11.7 18.2 16.9 9.3 14.9 15.1 14.6 12.8 15.8 16.8 9.9 10.1 11.2 16.4 Test whether the mean length of hospital stay is the same or different for the various strains of COVID-19. Construct the ANOVA table, find the p -value, and state your conclusion. Use a 5% significance level. We test for equality of mean length of hospital stay: H 0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 H a : μ i ≠ μ j , some i ≠ j The one-way ANOVA table results are shown: Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F P-value Factor (Between) 14.056 5 – 1 = 4 3.514 0.353095 0.8362 Error (Within) 99.52 15 – 5 = 10 9.952 Total 113.576 15 – 1 = 14 Distribution for the test: F 4 , 10 Probability Statement: p - v a l u e = P ( F > 0 . 3531 ) = 0 . 8362 Compare  and the p -value: α = 0 . 05 , p - v a l u e = 08362 , α > p - v a l u e Make a decision: Since  > p -value, we do not reject H 0 . Conclusion: At the 5% significance level, there is insufficient evidence from these data that different strains of COVID-19 virus will cause a significant difference in the length of a hospital stay. Four sororities took a random sample of sisters regarding their grade means for the past term. The results are shown in . Mean grades for four sororities Sorority 1 Sorority 2 Sorority 3 Sorority 4 2.17 2.63 2.63 3.79 1.85 1.77 3.78 3.45 2.83 3.25 4.00 3.08 1.69 1.86 2.55 2.26 3.33 2.21 2.45 3.18 Using a significance level of 1%, is there a difference in mean grades among the sororities? Let μ 1 , μ 2 , μ 3 , μ 4 be the population means of the sororities. Remember that the null hypothesis claims that the sorority groups are from the same normal distribution. The alternate hypothesis says that at least two of the sorority groups come from populations with different normal distributions. Notice that the four sample sizes are each five. NOTE This is an example of a balanced design , because each factor (i.e., sorority) has the same number of observations. H 0 : μ 1 = μ 2 = μ 3 = μ 4 H a : Not all of the means μ 1 , μ 2 , μ 3 , μ 4 are equal. Distribution for the test: F 3,16 where k = 4 groups and n = 20 samples in total df ( num )= k – 1 = 4 – 1 = 3 df ( denom ) = n – k = 20 – 4 = 16 Calculate the test statistic: F = 2.23 Graph: Probability statement: p -value = P ( F > 2.23) = 0.1241 Compare α and the p -value: α = 0.01 p -value = 0.1241 α < p -value Make a decision: Since α < p -value, you cannot reject H 0 . Conclusion: There is not sufficient evidence to conclude that there is a difference among the mean grades for the sororities. Try It Four sports teams took a random sample of players regarding their GPAs for the last year. The results are shown in . GPAs for four sports teams Basketball Baseball Hockey Lacrosse 3.6 2.1 4.0 2.0 2.9 2.6 2.0 3.6 2.5 3.9 2.6 3.9 3.3 3.1 3.2 2.7 3.8 3.4 3.2 2.5 Use a significance level of 5%, and determine if there is a difference in GPA among the teams. With a p -value of 0.9271, we decline to reject the null hypothesis. There is not sufficient evidence to conclude that there is a difference among the GPAs for the sports teams. A fourth grade class is studying the environment. One of the assignments is to grow bean plants in different soils. Tommy chose to grow his bean plants in soil found outside his classroom mixed with dryer lint. Tara chose to grow her bean plants in potting soil bought at the local nursery. Nick chose to grow his bean plants in soil from his mother's garden. No chemicals were used on the plants, only water. They were grown inside the classroom next to a large window. Each child grew five plants. At the end of the growing period, each plant was measured, producing the data (in inches) in . Tommy's plants Tara's plants Nick's plants 24 25 23 21 31 27 23 23 22 30 20 30 23 28 20 Does it appear that the three media in which the bean plants were grown produce the same mean height? Test at a 3% level of significance. This time, we will perform the calculations that lead to the F' statistic. Notice that each group has the same number of plants, so we will use the formula F' = n ⋅ s x – 2 s 2 pooled . First, calculate the sample mean and sample variance of each group. Tommy's plants Tara's plants Nick's plants Sample mean 24.2 25.4 24.4 Sample variance 11.7 18.3 16.3 Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4). Variance of the group means = 0.413 = s x – 2 Then MS between = n s x – 2 = (5)(0.413) where n = 5 is the sample size (number of plants each child grew). Calculate the mean of the three sample variances (Calculate the mean of 11.7, 18.3, and 16.3). Mean of the sample variances = 15.433 = s 2 pooled Then MS within = s 2 pooled = 15.433. The F statistic (or F ratio) is F = M S between M S within = n s x – 2 s 2 p o o l e d = ( 5 ) ( 0.413 ) 15.433 = 0.134 The dfs for the numerator = the number of groups – 1 = 3 – 1 = 2. The dfs for the denominator = the total number of samples – the number of groups = 15 – 3 = 12 The distribution for the test is F 2,12 and the F statistic is F = 0.134 The p -value is P ( F > 0.134) = 0.8759. Decision: Since α = 0.03 and the p -value = 0.8759, then you cannot reject H 0 . (Why?) Conclusion: With a 3% level of significance, from the sample data, the evidence is not sufficient to conclude that the mean heights of the bean plants are different. Try It Another fourth grader also grew bean plants, but this time in a jelly-like mass. The heights were (in inches) 24, 28, 25, 30, and 32. Do a one-way ANOVA test on the four groups. Are the heights of the bean plants different? Use the same method as shown in . Notation The notation for the F distribution is F ~ F df ( num ), df ( denom ) where df ( num ) = df between and df ( denom ) = df within The mean for the F distribution is μ = d f ( n u m ) d f ( d e n o m ) – 2 References Tomato Data, Marist College School of Science (unpublished student research) Chapter Review Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is the F statistic from an F distribution with (number of groups – 1) as the numerator degrees of freedom and (number of observations – number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table. Formula Review S S between = ∑  [ ( s j ) 2 n j ] − ( ∑  s j ) 2 n S S total = ∑  x 2 − ( ∑  x ) 2 n S S within = S S total − S S between df between = df ( num ) = k – 1 df within = df(denom) = n – k MS between = S S between d f between MS within = S S within d f within F = M S between M S within k = the number of groups n j = the size of the j th group s j = the sum of the values in the j th group n = the total number of all values (observations) combined x = one value (one observation) from the data s x – 2 = the variance of the sample means s 2 p o o l e d = the mean of the sample variances (pooled variance) Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in are the weights for the different groups. Group 1 Group 2 Group 3 216 202 170 198 213 165 240 284 182 187 228 197 176 210 201 What is the Sum of Squares Factor? 4,939.2 What is the Sum of Squares Error? What is the df for the numerator? 2 What is the df for the denominator? What is the Mean Square Factor? 2,469.6 What is the Mean Square Error? What is the F statistic? 3.7416 Use the following information to answer the next eight exercises. Players from four different soccer teams are to be tested for mean goals scored per game. The entries in are the goals per game for the different teams. Team 1 Team 2 Team 3 Team 4 1 2 0 3 2 3 1 4 0 2 1 4 3 4 0 3 2 4 0 2 What is SS between ? What is the df for the numerator? 3 What is MS between ? What is SS within ? 13.2 What is the df for the denominator? What is MS within ? 0.825 What is the F statistic? Judging by the F statistic, do you think it is likely or unlikely that you will reject the null hypothesis? Because a one-way ANOVA test is always right-tailed, a high F statistic corresponds to a low p -value, so it is likely that we cannot accept the null hypothesis. Homework Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. Northeast South West Central East 16.3 16.9 16.4 16.2 17.1 16.1 16.5 16.5 16.6 17.2 16.4 16.4 16.6 16.5 16.6 16.5 16.2 16.1 16.4 16.8 x – = ________ ________ ________ ________ ________ s 2 = ________ ________ ________ ________ ________ H 0 : µ 1 = µ 2 = µ 3 = µ 4 = µ 5 Hα : At least any two of the group means µ 1 , µ 2 , …, µ 5 are not equal. degrees of freedom – numerator: df ( num ) = _________ degrees of freedom – denominator: df ( denom ) = ________ df ( denom ) = 15 F statistic = ________

Business


Facts About the F Distribution Here are some facts about the F distribution. The curve is not symmetrical but skewed to the right. There is a different curve for each set of degrees of freedom. The F statistic is greater than or equal to zero. As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal as can be seen in the two figures below. Figure (b) with more degrees of freedom is more closely approaching the normal distribution, but remember that the F cannot ever be less than zero so the distribution does not have a tail that goes to infinity on the left as the normal distribution does. Other uses for the F distribution include comparing two variances and two-way Analysis of Variance. Two-Way Analysis is beyond the scope of this chapter. References Data from a fourth grade classroom in 1994 in a private K – 12 school in San Jose, CA. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall, 1994. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 50. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 118. “MLB Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012. Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich," Journal of the American Medical Association , 268, 1578-1580. Chapter Review The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The F statistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representing a large p -value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large F statistic and a small area (small p -value) to the right of the statistic under the F curve. When the data have unequal group sizes (unbalanced data), then techniques from need to be used for hand calculations. In the case of balanced data (the groups are the same size) however, simplified calculations based on group means and variances may be used. In practice, of course, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look at your data! An F statistic can have what values? What happens to the curves as the degrees of freedom for the numerator and the denominator get larger? The curves approximate the normal distribution. Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in . Team 1 Team 2 Team 3 Team 4 Team 5 36 32 48 38 41 42 35 50 44 39 51 38 39 46 40 What is the df(num) ? What is the df(denom) ? ten What are the Sum of Squares and Mean Squares Factors? What are the Sum of Squares and Mean Squares Errors? SS = 237.33; MS = 23.73 What is the F statistic? What is the p -value? 0.1614 At the 5% significance level, is there a difference in the mean jump heights among the teams? Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Group A Group B Group C 101 151 101 108 149 109 98 160 198 107 112 186 111 126 160 What is the df(num) ? two What is the df(denom) ? What are the SS between and MS between ? SS = 5,700.4; MS = 2,850.2 What are the SS within and MS within ? What is the F Statistic? 3.6101 What is the p -value? At the 10% significance level, are the scores among the different groups different? Yes, there is enough evidence to show that the scores among the groups are statistically significant at the 10% level. Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. Northeast South West Central East 16.3 16.9 16.4 16.2 17.1 16.1 16.5 16.5 16.6 17.2 16.4 16.4 16.6 16.5 16.6 16.5 16.2 16.1 16.4 16.8 x ¯ = ________ ________ ________ ________ ________ s 2 = ________ ________ ________ ________ ________ Enter the data into your calculator or computer. p -value = ______ State the decisions and conclusions (in complete sentences) for the following preconceived levels of α . α = 0.05 a. Decision: ____________________________ b. Conclusion: ____________________________ α = 0.01 a. Decision: ____________________________ b. Conclusion: ____________________________ Homework Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat's weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again, and the net gain in grams is recorded. Using a significance level of 10%, test the hypothesis that the three formulas produce the same mean weight gain. Weights of Student Lab Rats Linda's rats Tuan's rats Javier's rats 43.5 47.0 51.2 39.4 40.5 40.9 41.3 38.9 37.9 46.0 46.3 45.0 38.2 44.2 48.6 H 0 : µ L = µ T = µ J H a : at least any two of the means are different df ( num ) = 2; df ( denom ) = 12 F distribution 0.67 0.5305 Answers may vary. Decision: Cannot reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the means are different. A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are in . Using a 5% significance level, test the hypothesis that the three mean commuting mileages are the same. Working-class Professional (middle incomes) Professional (wealthy) 17.8 16.5 8.5 26.7 17.4 6.3 49.4 22.0 4.6 9.4 7.4 12.6 65.4 9.4 11.0 47.1 2.1 28.6 19.5 6.4 15.4 51.2 13.9 9.3 Use the following information to answer the next two exercises. lists the number of pages in four different types of magazines. Home decorating News Health Computer 172 87 82 104 286 94 153 136 163 123 87 98 205 106 103 207 197 101 96 146 Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length. Eliminate one magazine type that you now feel has a mean length different from the others. Redo the hypothesis test, testing that the remaining three means are statistically the same. Use a new solution sheet. Based on this test, are the mean lengths for the remaining three magazines statistically the same? H a : µ c = µ n = µ h At least any two of the magazines have different mean lengths. df ( num ) = 2, df ( denom ) = 12 F distribution F = 15.28 p -value = 0.001 Answers may vary. Alpha: 0.05 Decision: Cannot accept the null hypothesis. Reason for decision: p -value < alpha Conclusion: There is sufficient evidence to conclude that the mean lengths of the magazines are different. A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that shows the results of a study. CNN FOX Local 45 15 72 12 43 37 18 68 56 38 50 60 23 31 51 35 22 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. Are the means for the final exams the same for all statistics class delivery types? shows the scores on final exams from several randomly selected classes that used the different delivery types. Online Hybrid Face-to-Face 72 83 80 84 73 78 77 84 84 80 81 81 81 86 79 82 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. H 0 : μ o = μ h = μ f At least two of the means are different. df ( n ) = 2, df ( d ) = 13 F 2,13 0.64 0.5437 Answers may vary. Alpha: 0.05 Decision: Cannot reject the null hypothesis. Reason for decision: p -value > alpha Conclusion: The mean scores of different class delivery are not different. Are the mean number of times a month a person eats out the same for White people, Black people, Hispanic/Latinos and Asian people? Suppose that shows the results of a study. White Black Hispanic/Latino Asian 6 4 7 8 8 1 3 3 2 5 5 5 4 2 4 1 6 6 7 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. Are the mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that shows the results of a study. Powder Machine Made Hard Packed 1,210 2,107 2,846 1,080 1,149 1,638 1,537 862 2,019 941 1,870 1,178 1,528 2,233 1,382 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. H 0 : μ p = μ m = μ h At least any two of the means are different. df ( n ) = 2, df ( d ) = 12 F 2,12 3.13 0.0807 Answers may vary. Alpha: 0.05 Decision: Cannot reject the null hypothesis. Reason for decision: p -value > alpha Conclusion: There is not sufficient evidence to conclude that the mean numbers of daily visitors are different. Sanjay made identical paper airplanes out of three different weights of paper, light, medium and heavy. He made four airplanes from each of the weights, and launched them himself across the room. Here are the distances (in meters) that his planes flew. Paper type/Trial Trial 1 Trial 2 Trial 3 Trial 4 Heavy 5.1 meters 3.1 meters 4.7 meters 5.3 meters Medium 4 meters 3.5 meters 4.5 meters 6.1 meters Light 3.1 meters 3.3 meters 2.1 meters 1.9 meters Take a look at the data in the graph. Look at the spread of data for each group (light, medium, heavy). Does it seem reasonable to assume a normal distribution with the same variance for each group? Yes or No. Why is this a balanced design? Calculate the sample mean and sample standard deviation for each group. Does the weight of the paper have an effect on how far the plane will travel? Use a 1% level of significance. Complete the test using the method shown in the bean plant example in . variance of the group means __________ MS between = ___________ mean of the three sample variances ___________ MS within = _____________ F statistic = ____________ df(num) = __________, df(denom) = ___________ number of groups _______ number of observations _______ p -value = __________ ( P ( F > _______) = __________) Graph the p -value. decision: _______________________ conclusion: _______________________________________________________________ DDT is a pesticide that has been banned from use in the United States and most other areas of the world. It is quite effective, but persisted in the environment and over time became seen as harmful to higher-level organisms. Famously, egg shells of eagles and other raptors were believed to be thinner and prone to breakage in the nest because of ingestion of DDT in the food chain of the birds. An experiment was conducted on the number of eggs (fecundity) laid by female fruit flies. There are three groups of flies. One group was bred to be resistant to DDT (the RS group). Another was bred to be especially susceptible to DDT (SS). Finally there was a control line of non-selected or typical fruitflies (NS). Here are the data: RS SS NS RS SS NS 12.8 38.4 35.4 22.4 23.1 22.6 21.6 32.9 27.4 27.5 29.4 40.4 14.8 48.5 19.3 20.3 16 34.4 23.1 20.9 41.8 38.7 20.1 30.4 34.6 11.6 20.3 26.4 23.3 14.9 19.7 22.3 37.6 23.7 22.9 51.8 22.6 30.2 36.9 26.1 22.5 33.8 29.6 33.4 37.3 29.5 15.1 37.9 16.4 26.7 28.2 38.6 31 29.5 20.3 39 23.4 44.4 16.9 42.4 29.3 12.8 33.7 23.2 16.1 36.6 14.9 14.6 29.2 23.6 10.8 47.4 27.3 12.2 41.7 The values are the average number of eggs laid daily for each of 75 flies (25 in each group) over the first 14 days of their lives. Using a 1% level of significance, are the mean rates of egg selection for the three strains of fruitfly different? If so, in what way? Specifically, the researchers were interested in whether or not the selectively bred strains were different from the nonselected line, and whether the two selected lines were different from each other. Here is a chart of the three groups: The data appear normally distributed from the chart and of similar spread. There do not appear to be any serious outliers, so we may proceed with our ANOVA calculations, to see if we have good evidence of a difference between the three groups. H 0 : µ 1 = µ 2 = µ 3 ; H a : µ i ≠ ; some i ≠ j Define μ 1 , μ 2 , μ 3 , as the population mean number of eggs laid by the three groups of fruit flies. F statistic = 8.6657; p -value = 0.0004 Decision: Since the p -value is less than the level of significance of 0.01, we reject the null hypothesis. Conclusion: We have good evidence that the average number of eggs laid during the first 14 days of life for these three strains of fruitflies are different. Interestingly, if you perform a two sample t -test to compare the RS and NS groups they are significantly different ( p = 0.0013). Similarly, SS and NS are significantly different ( p = 0.0006). However, the two selected groups, RS and SS are not significantly different ( p = 0.5176). Thus we appear to have good evidence that selection either for resistance or for susceptibility involves a reduced rate of egg production (for these specific strains) as compared to flies that were not selected for resistance or susceptibility to DDT. Here, genetic selection has apparently involved a loss of fecundity. The data shown is the recorded body temperatures of 130 subjects as estimated from available histograms. Traditionally we are taught that the normal human body temperature is 98.6 F. This is not quite correct for everyone. Are the mean temperatures among the four groups different? Calculate 95% confidence intervals for the mean body temperature in each group and comment about the confidence intervals. FL FH ML MH FL FH ML MH 96.4 96.8 96.3 96.9 98.4 98.6 98.1 98.6 96.7 97.7 96.7 97 98.7 98.6 98.1 98.6 97.2 97.8 97.1 97.1 98.7 98.6 98.2 98.7 97.2 97.9 97.2 97.1 98.7 98.7 98.2 98.8 97.4 98 97.3 97.4 98.7 98.7 98.2 98.8 97.6 98 97.4 97.5 98.8 98.8 98.2 98.8 97.7 98 97.4 97.6 98.8 98.8 98.3 98.9 97.8 98 97.4 97.7 98.8 98.8 98.4 99 97.8 98.1 97.5 97.8 98.8 98.9 98.4 99 97.9 98.3 97.6 97.9 99.2 99 98.5 99 97.9 98.3 97.6 98 99.3 99 98.5 99.2 98 98.3 97.8 98 99.1 98.6 99.5 98.2 98.4 97.8 98 99.1 98.6 98.2 98.4 97.8 98.3 99.2 98.7 98.2 98.4 97.9 98.4 99.4 99.1 98.2 98.4 98 98.4 99.9 99.3 98.2 98.5 98 98.6 100 99.4 98.2 98.6 98 98.6 100.8

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Introduction Linear regression and correlation can help you determine if an auto mechanic’s salary is related to his work experience. (credit: modification of work “USPS commissions local repair-shop for some needed work on its older trucks” by Joshua Rothhaas/ Flickr, CC BY 2.0) Professionals often want to know how two or more numeric variables are related. For example, is there a relationship between the grade on the second math exam a student takes and the grade on the final exam? If there is a relationship, what is the relationship and how strong is it? In another example, your income may be determined by your education, your profession, your years of experience, and your ability, or your gender or color. The amount you pay a repair person for labor is often determined by an initial amount plus an hourly fee. These examples may or may not be tied to a model, meaning that some theory suggested that a relationship exists. This link between a cause and an effect, often referred to as a model, is the foundation of the scientific method and is the core of how we determine what we believe about how the world works. Beginning with a theory and developing a model of the theoretical relationship should result in a prediction, what we have called a hypothesis earlier. Now the hypothesis concerns a full set of relationships. As an example, in Economics the model of consumer choice is based upon assumptions concerning human behavior: a desire to maximize something called utility, knowledge about the benefits of one product over another, likes and dislikes, referred to generally as preferences, and so on. These combined to give us the demand curve. From that we have the prediction that as prices rise the quantity demanded will fall. Economics has models concerning the relationship between what prices are charged for goods and the market structure in which the firm operates, monopoly verse competition, for example. Models for who would be most likely to be chosen for an on-the-job training position, the impacts of Federal Reserve policy changes and the growth of the economy and on and on. Models are not unique to Economics, even within the social sciences. In political science, for example, there are models that predict behavior of bureaucrats to various changes in circumstances based upon assumptions of the goals of the bureaucrats. There are models of political behavior dealing with strategic decision making both for international relations and domestic politics. The so-called hard sciences are, of course, the source of the scientific method as they tried through the centuries to explain the confusing world around us. Some early models today make us laugh; spontaneous generation of life for example. These early models are seen today as not much more than the foundational myths we developed to help us bring some sense of order to what seemed chaos. The foundation of all model building is the perhaps the arrogant statement that we know what caused the result we see. This is embodied in the simple mathematical statement of the functional form that y = f(x). The response, Y, is caused by the stimulus, X. Every model will eventually come to this final place and it will be here that the theory will live or die. Will the data support this hypothesis? If so then fine, we shall believe this version of the world until a better theory comes to replace it. This is the process by which we moved from flat earth to round earth, from earth-center solar system to sun-center solar system, and on and on. The scientific method does not confirm a theory for all time: it does not prove “truth”. All theories are subject to review and may be overturned. These are lessons we learned as we first developed the concept of the hypothesis test earlier in this book. Here, as we begin this section, these concepts deserve review because the tool we will develop here is the cornerstone of the scientific method and the stakes are higher. Full theories will rise or fall because of this statistical tool; regression and the more advanced versions call econometrics. In this chapter we will begin with correlation, the investigation of relationships among variables that may or may not be founded on a cause and effect model. The variables simply move in the same, or opposite, direction. That is to say, they do not move randomly. Correlation provides a measure of the degree to which this is true. From there we develop a tool to measure cause and effect relationships; regression analysis. We will be able to formulate models and tests to determine if they are statistically sound. If they are found to be so, then we can use them to make predictions: if as a matter of policy we changed the value of this variable what would happen to this other variable? If we imposed a gasoline tax of 50 cents per gallon how would that effect the carbon emissions, sales of Hummers/Hybrids, use of mass transit, etc.? The ability to provide answers to these types of questions is the value of regression as both a tool to help us understand our world and to make thoughtful policy decisions.

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The Correlation Coefficient r As we begin this section we note that the type of data we will be working with has changed. Perhaps unnoticed, all the data we have been using is for a single variable. It may be from two samples, but it is still a univariate variable. The type of data described in the examples above and for any model of cause and effect is bivariate data — "bi" for two variables. In reality, statisticians use multivariate data, meaning many variables. For our work we can classify data into three broad categories, time series data, cross-section data, and panel data. We met the first two very early on. Time series data measures a single unit of observation; say a person, or a company or a country, as time passes. What are measured will be at least two characteristics, say the person’s income, the quantity of a particular good they buy and the price they paid. This would be three pieces of information in one time period, say 1985. If we followed that person across time we would have those same pieces of information for 1985,1986, 1987, etc. This would constitute a times series data set. If we did this for 10 years we would have 30 pieces of information concerning this person’s consumption habits of this good for the past decade and we would know their income and the price they paid. A second type of data set is for cross-section data. Here the variation is not across time for a single unit of observation, but across units of observation during one point in time. For a particular period of time we would gather the price paid, amount purchased, and income of many individual people. A third type of data set is panel data. Here a panel of units of observation is followed across time. If we take our example from above we might follow 500 people, the unit of observation, through time, ten years, and observe their income, price paid and quantity of the good purchased. If we had 500 people and data for ten years for price, income and quantity purchased we would have 15,000 pieces of information. These types of data sets are very expensive to construct and maintain. They do, however, provide a tremendous amount of information that can be used to answer very important questions. As an example, what is the effect on the labor force participation rate of women as their family of origin, mother and father, age? Or are there differential effects on health outcomes depending upon the age at which a person started smoking? Only panel data can give answers to these and related questions because we must follow multiple people across time. The work we do here however will not be fully appropriate for data sets such as these. Beginning with a set of data with two independent variables we ask the question: are these related? One way to visually answer this question is to create a scatter plot of the data. We could not do that before when we were doing descriptive statistics because those data were univariate. Now we have bivariate data so we can plot in two dimensions. Three dimensions are possible on a flat piece of paper, but become very hard to fully conceptualize. Of course, more than three dimensions cannot be graphed although the relationships can be measured mathematically. To provide mathematical precision to the measurement of what we see we use the correlation coefficient. The correlation tells us something about the co-movement of two variables, but nothing about why this movement occurred. Formally, correlation analysis assumes that both variables being analyzed are independent variables. This means that neither one causes the movement in the other. Further, it means that neither variable is dependent on the other, or for that matter, on any other variable. Even with these limitations, correlation analysis can yield some interesting results. The correlation coefficient , ρ (pronounced rho), is the mathematical statistic for a population that provides us with a measurement of the strength of a linear relationship between the two variables. For a sample of data, the statistic, r, developed by Karl Pearson in the early 1900s, is an estimate of the population correlation and is defined mathematically as: r = 1 n − 1 Σ ( X 1 i − X ¯ 1 ) ( X 2 i − X ¯ 2 ) s x 1 s x 2 OR r = ∑ ( X 1 i - X ¯ 1 ) ( X 2 i - X ¯ 2 ) ∑ ( X 1 i - X ¯ 1 ) 2 ∑ ( X 2 i - X ¯ 2 ) 2 where s x1 and s x2 are the standard deviations of the two independent variables X 1 and X 2 , X ¯ 1 and X ¯ 2 are the sample means of the two variables, and X 1i and X 2i are the individual observations of X 1 and X 2 . The correlation coefficient r ranges in value from -1 to 1. The second equivalent formula is often used because it may be computationally easier. As scary as these formulas look they are really just the ratio of the covariance between the two variables and the product of their two standard deviations. That is to say, it is a measure of relative variances. In practice all correlation and regression analysis will be provided through computer software designed for these purposes. Anything more than perhaps one-half a dozen observations creates immense computational problems. It was because of this fact that correlation, and even more so, regression, were not widely used research tools until after the advent of “computing machines”. Now the computing power required to analyze data using regression packages is deemed almost trivial by comparison to just a decade ago. To visualize any linear relationship that may exist review the plot of a scatter diagrams of the standardized data. presents several scatter diagrams and the calculated value of r. In panels (a) and (b) notice that the data generally trend together, (a) upward and (b) downward. Panel (a) is an example of a positive correlation and panel (b) is an example of a negative correlation, or relationship. The sign of the correlation coefficient tells us if the relationship is a positive or negative (inverse) one. If all the values of X 1 and X 2 are on a straight line the correlation coefficient will be either 1 or -1 depending on whether the line has a positive or negative slope and the closer to one or negative one the stronger the relationship between the two variables. BUT ALWAYS REMEMBER THAT THE CORRELATION COEFFICIENT DOES NOT TELL US THE SLOPE. Remember, all the correlation coefficient tells us is whether or not the data are linearly related. In panel (d) the variables obviously have some type of very specific relationship to each other, but the correlation coefficient is zero, indicating no linear relationship exists. If you suspect a linear relationship between X 1 and X 2 then r can measure how strong the linear relationship is. What the VALUE of r tells us: The value of r is always between –1 and +1: –1 ≤ r ≤ 1. The size of the correlation r indicates the strength of the linear relationship between X 1 and X 2 . Values of r close to –1 or to +1 indicate a stronger linear relationship between X 1 and X 2 . If r = 0 there is absolutely no linear relationship between X 1 and X 2 (no linear correlation) . If r = 1, there is perfect positive correlation. If r = –1, there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line: ANY straight line no matter what the slope. Of course, in the real world, this will not generally happen. What the SIGN of r tells us A positive value of r means that when X 1 increases, X 2 tends to increase and when X 1 decreases, X 2 tends to decrease (positive correlation) . A negative value of r means that when X 1 increases, X 2 tends to decrease and when X 1 decreases, X 2 tends to increase (negative correlation) . NOTE Strong correlation does not suggest that X 1 causes X 2 or X 2 causes X 1 . We say "correlation does not imply causation." In order to have a correlation coefficient between traits A and B, it is necessary to have: one group of subjects, some of whom possess characteristics of trait A, the remainder possessing those of trait B measures of trait A on one group of subjects and of trait B on another group two groups of subjects, one which could be classified as A or not A, the other as B or not B two groups of subjects, one which could be classified as A or not A, the other as B or not B d Define the Correlation Coefficient and give a unique example of its use. A measure of the degree to which variation of one variable is related to variation in one or more other variables. The most commonly used correlation coefficient indicates the degree to which variation in one variable is described by a straight line relation with another variable. Suppose that sample information is available on family income and Years of schooling of the head of the household. A correlation coefficient = 0 would indicate no linear association at all between these two variables. A correlation of 1 would indicate perfect linear association (where all variation in family income could be associated with schooling and vice versa). If the correlation between age of an auto and money spent for repairs is +.90 81% of the variation in the money spent for repairs is explained by the age of the auto 81% of money spent for repairs is unexplained by the age of the auto 90% of the money spent for repairs is explained by the age of the auto none of the above a. 81% of the variation in the money spent for repairs is explained by the age of the auto Suppose that college grade-point average and verbal portion of an IQ test had a correlation of .40. What percentage of the variance do these two have in common? 20 16 40 80 b. 16 True or false? If false, explain why: The coefficient of determination can have values between -1 and +1. The coefficient of determination is r··2 with 0 ≤ r··2 ≤ 1, since -1 ≤ r ≤ 1. True or False: Whenever r is calculated on the basis of a sample, the value which we obtain for r is only an estimate of the true correlation coefficient which we would obtain if we calculated it for the entire population. True Under a "scatter diagram" there is a notation that the coefficient of correlation is .10. What does this mean? plus and minus 10% from the means includes about 68% of the cases one-tenth of the variance of one variable is shared with the other variable one-tenth of one variable is caused by the other variable on a scale from -1 to +1, the degree of linear relationship between the two variables is +.10 d. on a scale from -1 to +1, the degree of linear relationship between the two variables is +.10 The correlation coefficient for X and Y is known to be zero. We then can conclude that: X and Y have standard distributions the variances of X and Y are equal there exists no relationship between X and Y there exists no linear relationship between X and Y none of these d. there exists no linear relationship between X and Y What would you guess the value of the correlation coefficient to be for the pair of variables: "number of man-hours worked" and "number of units of work completed"? Approximately 0.9 Approximately 0.4 Approximately 0.0 Approximately -0.4 Approximately -0.9 Approximately 0.9 In a given group, the correlation between height measured in feet and weight measured in pounds is +.68. Which of the following would alter the value of r? height is expressed centimeters. weight is expressed in Kilograms. both of the above will affect r. neither of the above changes will affect r. d. neither of the above changes will affect r. Bivariate two variables are present in the model where one is the “cause” or independent variable and the other is the “effect” of dependent variable. Multivariate a system or model where more than one independent variable is being used to predict an outcome. There can only ever be one dependent variable, but there is no limit to the number of independent variables. R – Correlation Coefficient A number between −1 and 1 that represents the strength and direction of the relationship between “X” and “Y.” The value for “ r ” will equal 1 or −1 only if all the plotted points form a perfectly straight line. Linear a model that takes data and regresses it into a straight line equation.

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Testing the Significance of the Correlation Coefficient The correlation coefficient, r , tells us about the strength and direction of the linear relationship between X 1 and X 2 . The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient. ρ = population correlation coefficient (unknown) r = sample correlation coefficient (known; calculated from sample data) The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient r and the sample size n . If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant." Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between X 1 and X 2 because the correlation coefficient is significantly different from zero. What the conclusion means: There is a significant linear relationship X 1 and X 2 . If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant". Performing the Hypothesis Test Null Hypothesis: H 0 : ρ = 0 Alternate Hypothesis: H a : ρ ≠ 0 What the Hypotheses Mean in Words Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship (correlation) between X 1 and X 2 in the population. Alternate Hypothesis H a : The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between X 1 and X 2 in the population. Drawing a Conclusion There are two methods of making the decision concerning the hypothesis. The test statistic to test this hypothesis is: t c = r ( 1 − r 2 ) ( n − 2 ) OR t c = r n − 2 1 − r 2 Where the second formula is an equivalent form of the test statistic, n is the sample size and the degrees of freedom are n-2. This is a t-statistic and operates in the same way as other t tests. Calculate the t-value and compare that with the critical value from the t-table at the appropriate degrees of freedom and the level of confidence you wish to maintain. If the calculated value is in the tail then cannot accept the null hypothesis that there is no linear relationship between these two independent random variables. If the calculated t-value is NOT in the tailed then cannot reject the null hypothesis that there is no linear relationship between the two variables. A quick shorthand way to test correlations is the relationship between the sample size and the correlation. If: | r | ≥ 2 n then this implies that the correlation between the two variables demonstrates that a linear relationship exists and is statistically significant at approximately the 0.05 level of significance. As the formula indicates, there is an inverse relationship between the sample size and the required correlation for significance of a linear relationship. With only 10 observations, the required correlation for significance is 0.6325, for 30 observations the required correlation for significance decreases to 0.3651 and at 100 observations the required level is only 0.2000. Correlations may be helpful in visualizing the data, but are not appropriately used to "explain" a relationship between two variables. Perhaps no single statistic is more misused than the correlation coefficient. Citing correlations between health conditions and everything from place of residence to eye color have the effect of implying a cause and effect relationship. This simply cannot be accomplished with a correlation coefficient. The correlation coefficient is, of course, innocent of this misinterpretation. It is the duty of the analyst to use a statistic that is designed to test for cause and effect relationships and report only those results if they are intending to make such a claim. The problem is that passing this more rigorous test is difficult so lazy and/or unscrupulous "researchers" fall back on correlations when they cannot make their case legitimately. Define a t Test of a Regression Coefficient, and give a unique example of its use. Definition: A t test is obtained by dividing a regression coefficient by its standard error and then comparing the result to critical values for Students' t with Error df . It provides a test of the claim that β i = 0 when all other variables have been included in the relevant regression model. Example: Suppose that 4 variables are suspected of influencing some response. Suppose that the results of fitting Y i = β 0 + β 1 X 1 i + β 2 X 2 i + β 3 X 3 i + β 4 X 4 i + e i include: Variable Regression coefficient Standard error of regular coefficient .5 1 -3 .4 2 +2 .02 3 +1 .6 4 -.5 t calculated for variables 1, 2, and 3 would be 5 or larger in absolute value while that for variable 4 would be less than 1. For most significance levels, the hypothesis β 1 = 0 would be rejected. But, notice that this is for the case when X 2 , X 3 , and X 4 have been included in the regression. For most significance levels, the hypothesis β 4 = 0 would be continued (retained) for the case where X 1 , X 2 , and X 3 are in the regression. Often this pattern of results will result in computing another regression involving only X 1 , X 2 , X 3 , and examination of the t ratios produced for that case. The correlation between scores on a neuroticism test and scores on an anxiety test is high and positive; therefore anxiety causes neuroticism those who score low on one test tend to score high on the other. those who score low on one test tend to score low on the other. no prediction from one test to the other can be meaningfully made. c. those who score low on one test tend to score low on the other.

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Linear Equations Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form: y = a + bx where a and b are constant numbers. The variable x is the independent variable, and y is the dependent variable. Another way to think about this equation is a statement of cause and effect. The X variable is the cause and the Y variable is the hypothesized effect. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable. The following examples are linear equations. y = 3 + 2x y = –0.01 + 1.2x Try It Is the following an example of a linear equation? y = –0.125 – 3.5 x The graph of a linear equation of the form y = a + bx is a straight line . Any line that is not vertical can be described by this equation. Graph the equation y = –1 + 2 x . Try It Is the following an example of a linear equation? Why or why not? No, the graph is not a straight line; therefore, it is not a linear equation. A local small business completes federal tax returns for customers. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job. Find the equation that expresses the total cost in terms of the number of hours required to complete the job. Let x = the number of hours it takes to get the job done. Let y = the total cost to the customer. The $31.50 is a fixed cost. If it takes x hours to complete the job, then (32)( x ) is the cost of the tax return processing only. The total cost is: y = 31.50 + 32 x Try It Emma’s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class. Slope and Y -Intercept of a Linear Equation For the linear equation y = a + bx , b = slope and a = y -intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the y -intercept is the y coordinate of the point (0, a ) where the line crosses the y -axis. From calculus the slope is the first derivative of the function. For a linear function the slope is dy / dx = b where we can read the mathematical expression as "the change in y ( dy ) that results from a change in x ( dx ) = b * dx ". Three possible graphs of y = a + bx . (a) If b > 0, the line slopes upward to the right. (b) If b = 0, the line is horizontal. (c) If b < 0, the line slopes downward to the right. Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y = 25 + 15 x . What are the independent and dependent variables? What is the y -intercept and what is the slope? Interpret them using complete sentences. The independent variable ( x ) is the number of hours Svetlana tutors each session. The dependent variable ( y ) is the amount, in dollars, Svetlana earns for each session. The y -intercept is 25 ( a = 25). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when x = 0). The slope is 15 ( b = 15). For each session, Svetlana earns $15 for each hour she tutors. Try It Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is y = 25 + 20 x . What are the independent and dependent variables? What is the y -intercept and what is the slope? Interpret them using complete sentences. True or False? If False, correct it: Suppose a 95% confidence interval for the slope β of the straight line regression of Y on X is given by -3.5 < β < -0.5. Then a two-sided test of the hypothesis H 0 : β = −1 would result in rejection of H 0 at the 1% level of significance. False. Since H 0 : β = −1 would not be rejected at α = 0.05 , it would not be rejected at α = 0.01 . True or False: It is safer to interpret correlation coefficients as measures of association rather than causation because of the possibility of spurious correlation. True We are interested in finding the linear relation between the number of widgets purchased at one time and the cost per widget. The following data has been obtained: X: Number of widgets purchased – 1, 3, 6, 10, 15 Y: Cost per widget(in dollars) – 55, 52, 46, 32, 25 Suppose the regression line is y ^ = −2.5 x + 60 . We compute the average price per widget if 30 are purchased and observe which of the following? y ^ = 15 dollars ; obviously, we are mistaken; the prediction y ^ is actually +15 dollars. y ^ = 15 dollars , which seems reasonable judging by the data. y ^ = −15 dollars , which is obvious nonsense. The regression line must be incorrect. y ^ = −15 dollars , which is obvious nonsense. This reminds us that predicting Y outside the range of X values in our data is a very poor practice. d Discuss briefly the distinction between correlation and causality. Some variables seem to be related, so that knowing one variable's status allows us to predict the status of the other. This relationship can be measured and is called correlation. However, a high correlation between two variables in no way proves that a cause-and-effect relation exists between them. It is entirely possible that a third factor causes both variables to vary together. True or False: If r is close to + or -1, we shall say there is a strong correlation, with the tacit understanding that we are referring to a linear relationship and nothing else. True Chapter Review The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b , where m and b are constants, x is the independent variable, y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx , where a and b are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx , the constant b , called the coefficient, represents the slope . The constant a is called the y-intercept . The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable ( y ) changes for every one unit increase in the independent ( x ) variable, on average. The y -intercept is used to describe the dependent variable when the independent variable equals zero. Y – the dependent variable Also, using the letter “y” represents actual values while y ^ represents predicted or estimated values. Predicted values will come from plugging in observed “x” values into a linear model. X – the independent variable This will sometimes be referred to as the “predictor” variable, because these values were measured in order to determine what possible outcomes could be predicted. a is the symbol for the Y-Intercept Sometimes written as b 0 , because when writing the theoretical linear model β 0 is used to represent a coefficient for a population. b is the symbol for Slope The word coefficient will be used regularly for the slope, because it is a number that will always be next to the letter “x.” It will be written as b 1 when a sample is used, and β 1 will be used with a population or when writing the theoretical linear model.

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