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Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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- Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?A random sample of twelve students were chosen, and their midterm test score (y), as- signment score (x1), and missed classes (x2) were recorded as follows: Midterm Score, y Assignment Score, x1 Classes Missed, x2 85 74 76 90 85 87 94 98 81 91 76 74 65 50 55 65 55 70 65 70 55 70 50 55 5 7 5 2 6 3 2 5 4 3 1 4 (i) What is the fitted multiple linear regression equation of the form yˆ = b0 + b1x1 + b2x2? (ii) From part (i) above, estimate the midterm test score grade for a student who has an assignment score of 60 and missed 4 classes.The table below shows the number of state-registered automatic weapons and the murder rate for several Northwestern states, where xx is thousands of automatic weapons and yy is murders per 100,000 residents. xx 11.3 8.2 7.1 3.7 2.9 2.2 2.1 0.6 yy 13.9 10.7 10.3 7.2 6.5 5.6 5.5 4.6 Use your calculator to determine the equation of the regression line and write it in the y=ax+by=ax+b form. Round to 2 decimal places. According to this model, how many murders per 100,000 residents can be expected in a state with 4.6 thousand automatic weapons? Round to 3 decimal places. According to this model, how many murders per 100,000 residents can be expected in a state with 4.4 thousand automatic weapons? Round to 3 decimal places.
- The number of initial public offerings of stock issued in a 10-year period and the total proceeds of these offerings (in millions) are shown in the table. The equation of the regression line is y=47.271x+18,712.47. Issues, x Proceeds, y425 18,758468 29,602678 43,797478 31,661485 65,518396 65,33069 21,42764 11,847181 32,392157 27,561 Complete parts a and b. The number of initial public offerings of stock issued in a 10-year period and the total proceeds of these offerings (in millions) are shown in the table. The equation of the regression line is y=47.271x+18,712.47. Complete parts a and b. (a) Find the coefficient of determination and interpret the result. (Round to three decimal places as needed.) (b) Find the standard error of estimate se and interpret the result. (Round to three decimal places as needed.)(b)A random sample of twelve students were chosen, and their midterm test score ( y), assignment score (x1), and missed classes (x2) were recorded as follows: Midterm Score, y Assignment Score, x1 Classes Missed, x2 85 65 1 + Ω 74 50 7 76 55 5 90 65 2 85 55 6 87 70 3 94 65 2 98 70 5 81 55 4 91 70 3 76 50 1 74 55 4 (i)What is the fitted multiple linear regression equation of the form ˆy = b0 + b1x1 + b2x2? (ii)From part 6(b)(i) above, estimate the midterm test score grade for a student who has an assignment score of 60 and missed 4 classes. (iii)Assume that the data on ( y, x1, x2) above in 6(b) is inputted into SPSS, with the following results: ANOVA Source SS df MS F…(b)A random sample of twelve students were chosen, and their midterm test score ( y), assignment score (x1), and missed classes (x2) were recorded as follows: Midterm Score, y Assignment Score, x1 Classes Missed, x2 85 65 1 + Ω 74 50 7 76 55 5 90 65 2 85 55 6 87 70 3 94 65 2 98 70 5 81 55 4 91 70 3 76 50 1 74 55 4 (i)What is the fitted multiple linear regression equation of the form ˆy = b0 + b1x1 + b2x2? (ii)From part 6(b)(i) above, estimate the midterm test score grade for a student who has an assignment score of 60 and missed 4 classes. (iii)Assume that the data on ( y, x1, x2) above in 6(b) is inputted into SPSS, with the following results: ANOVA Source SS df MS F…
- (b)A random sample of twelve students were chosen, and their midterm test score ( y), assignment score (x1), and missed classes (x2) were recorded as follows: Midterm Score, y Assignment Score, x1 Classes Missed, x2 85 65 8 74 50 7 76 55 5 90 65 2 85 55 6 87 70 3 94 65 2 98 70 5 81 55 4 91 70 3 76 50 1 74 55 4 (i)What is the fitted multiple linear regression equation of the form ˆy = b0 + b1x1 + b2x2? (ii)From part 6(b)(i) above, estimate the midterm test score grade for a student who has an assignment score of 60 and missed 4 classes. (iii)Assume that the data on ( y, x1, x2) above in 6(b) is inputted into SPSS, with the following results: ANOVA Source SS df MS F…(b)A random sample of twelve students were chosen, and their midterm test score ( y), assignment score (x1), and missed classes (x2) were recorded as follows: Midterm Score, y Assignment Score, x1 Classes Missed, x2 85 65 8 74 50 7 76 55 5 90 65 2 85 55 6 87 70 3 94 65 2 98 70 5 81 55 4 91 70 3 76 50 1 74 55 4 (i)What is the fitted multiple linear regression equation of the form ˆy = b0 + b1x1 + b2x2? (ii)From part 6(b)(i) above, estimate the midterm test score grade for a student who has an assignment score of 60 and missed 4 classes. (iii)Assume that the data on ( y, x1, x2) above in 6(b) is inputted into SPSS, with the following results: ANOVA Source SS df MS F…Interpret the estimated regression coefficient corresponding to the Z variable. Data Salary Education Experience Sex 29.7985 15 3 1 21.8219 4 0 0 22.8978 4 0 0 22.0917 1 1 0 21.8993 5 0 0 22.4829 3 1 1 28.0772 15 0 0 y=salary 23.6292 6 1 1 x1=education level in schooling years 32.3595 0 15 1 x2=experience level in employment level 21.794 1 0 0 d=sex (1 for male,0 for female) 19.8762 3 0 0 Ln(Y) = alpha +beta1X1 +Beta2X2+ Beta3D +Beta4Z +e 21.0253 3 0 0 where z =X2D 24.6323 0 5 1 19.0247 0 0 0 18.8857 0 0 0 21.8552 1 0 0 24.2675 6 1 0 18.7931 0 0 0 18.9276 0 0 0 23.4441 5 1 1 20.8047 2 0 0 18.26 0 0 0 20.6726 0 2 1 21.7815 3 0 0…
- For these (x,y) pairs of data points: 1,5 3,7 4,6 5,8 7,9 Compute . Compute . What is the equation of the regression line?The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows: Give the following: a. linear regression line and equation b. computation of the coefficient of determination ?^2 c. Computation of the coefficient of correlation ? d. Estimate the final examination grade of a student who received a grade of 85 on the midterm report.The following table shows the annual number of PhD graduates in a country in various fields. NaturalSciences Engineering SocialSciences Education 1990 70 10 60 30 1995 130 40 100 50 2000 330 130 280 140 2005 490 370 460 210 2010 590 550 830 520 2012 690 590 1,000 900 (a) With x = the number of social science doctorates and y = the number of education doctorates, use technology to obtain the regression equation. (Round coefficients to three significant digits.) y(x) =