Correlation

Background Is being physically strong still important in today’s workplace? In our current high-tech world one might be inclined to think that only skills required for computer work such as reading, reasoning, abstract thinking, etc. are important for performing well in many of today’s jobs. There are still, however, a number of very important jobs that require, in addition to cognitive skills, a significant amount of strength to be able to perform at a high level. Take, for example, the job of a construction worker.

It takes a lot of strength to lift, position, and secure many building materials such as wood boards, metal bars, and cement blocks. In addition, the tools used in construction work are often heavy and require a lot of strength to control. When was the last time you tried to operate a jackhammer? There are many more jobs such as electrician and auto mechanic that also require strength. An interesting applied problem that arises is how to select the best candidates from among a group of applicants for physically demanding jobs.

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One obvious way might be to take them to a job site and have them demonstrate that they are strong enough to do the job. Unfortunately, this approach might be too time consuming if you are having to select a large number of people from a large applicant pool. Also, you risk injury to applicants who are not strong enough to do the job. A solution to this problem is to develop a measure of physical ability that is easy and quick to administer, does not risk injury, and is related to how well a person performs the actual job.

A study by Blakely, Quinones, and Jago (1995) published in the journal Personnel Psychology reports on the research results of just such a measure. That study, and this case study, looks at methods for determining if these strength tests are related to performance on the job. The principles and methods associated with this case study also apply to any number of variables other than strength and job performance. Case Study Objectives The purpose of this case study is to describe the logic behind the statistical principles and procedures listed below within the context of an applied problem.

After a thorough exploration of this case study and all associated links you should be familiar with the principles and be able to apply them to similar situations. In addition, you should be able to use one of several popular statistical packages to carry out the analyses described in this case study. The specific statistical principles associated with this study are: * scatterplots * covariance * correlation * linear regression * multiple regression Recommended Use

The principles presented in this case study have a number of issues associated with them. Therefore, we recommend that you take the time to explore the various links presented in this case study. Some of those links will take you to additional information regarding a particular statistical procedure or issue. Others will take you to simulations which demonstrate the principles or techniques being presented. These simulations give you hands-on experience with the various statistical procedures and principles.

Study Method and Procedure Study Participants The data presented in this case study were collected from 147 individuals working in physically demanding jobs including electricians, construction and maintenance workers, auto mechanics, and linemen. An analysis of the tasks performed in these jobs showed that a number of them required a substantial amount of strength to perform. Here are some examples of physically demanding tasks performed in some of these jobs. * Uses hand tools (wrenches, pliers, hammer) Carries equipment, tools, and other materials to and from job sites * Secures job site by laying out, constructing, and installing shoring, barricades, and industrial fencing * Excavates for landscaping, trenches, and job site Physical Strength Measures Two measures of strength were gathered from each participant. These included grip and arm strength. A piece of equipment known as the Jackson Evaluation System (JES) was used to collect the strength data. The JES can be configured to measure the strength of a number of muscle groups. In this study, grip strength and arm strength were measured.

For each of the tests, the participant was asked to exert as much force as they could for a period of 2 seconds. The equipment then measured the maximum force exerted in pounds (lbs. ). Since there is little to no movement along any joints, these measures are often referred to as isometric strength measures. To increase the accuracy of the measure, participants were asked to perform each test twice. The two scores were then averaged together for each of the two strength measures (grip and arm). Job Performance Two separate measures of job performance are presented in this case study.

First, the supervisors for each of the 147 study participants were asked to rate how well their employee(s) perform on the physical aspects of their jobs using a 60-point scale. Higher numbers indicate better performance on the physically demanding aspects of the job. Second, two work simulations were developed by using information collected from an analysis of the jobs as well as observations and measurements of individuals performing each of the jobs studied. The simulations required that the participant exert force on a simulated wrench while assuming a standing and a kneeling position.

The scores on each of the simulations were standardized and summed together to form one single measure. Larger scores indicate better performance on the work simulations. Descriptive Statistics The first step in examining the relationship between strength and job performance is to look at some basic descriptive statistics for each of the four variables. Measures of central tendency such as the mean and the median can be used to determine the location of the distribution of scores. Measures of dispersion such as the range (minimum, maximum), the standard deviation, and the variance tell you how spread out the scores were.

Arm For the ARM test, we can see that the 147 study participants exerted an average of 78. 75 pounds of force. Half of the participants exerted less than 81. 50 pounds of force while the other half scored greater than 81. 50. The scores were fairly spread out with a standard deviation of 21. 11. The lowest score was 19 lbs. Someone managed to exert a maximum score of 132 lbs. N Mean Median Trimean Minimum Maximum 25th percentile 75th percentile s se of mean Skew Kurtosis| 147 78. 75 81. 50 80. 38 19. 00 132. 00 64. 50 94. 00 21. 11 1. 74 -0. 30 0. 01|

Another important step in evaluating a set of data is to look at the overall shape of the distributions. One way of doing that is to construct a stem-and-leaf graph like the one below. How many people exerted a force of 85 pounds? 2 3 4 5 Another good way to portray the shape of the distribution is with a histogram as shown below. Descriptive statistics and histograms for the remaining variables are presented below. Grip N Mean Median Trimean Minimum Maximum 25th percentile 75th percentile s se of mean Skew Kurtosis| 147 110. 23 111. 00 110. 12 29. 00 189. 00 94. 00 124. 50 23. 63 1. 95 0. 02 1. 4| If the distribution were normal, between which two values would 68% of the values lie? [0,1] [109,111] [85, 136] [90, 131] [87, 134] Ratings N Mean Median Trimean Minimum Maximum 25th percentile 75th percentile s se of mean Skew Kurtosis| 147 41. 01 41. 30 41. 30 21. 60 57. 20 34. 80 47. 80 8. 52 0. 70 -0. 11 -0. 84| The ratings variable: appears to deviate from normality. appears to approximately normally distributed. SIMS N Mean Median Trimean Minimum Maximum 25th percentile 75th percentile s se of mean Skew Kurtosis| 147 0. 20 0. 16 0. 11 -4. 17 5. 17 -0. 99 1. 11 1. 68 0. 14 0. 45 0. 68|

Notice the distribution has a slight positive skew (skewed to the right). Scatterplots Perhaps the most important step in examining the relationship between two variables is to create a scatterplot. A scatterplot is simply a graph which plots an individuals’ score on one variable (e. g. arm strength) against their score on a second variable (e. g. supervisory ratings). Scatterplots are used to examine any general trends in the relationship between two variables. If scores on one variable tend to increase with correspondingly high scores of the second variable, a positive relationship is said to exist.

If high scores on one variable are associated with low scores on the other, a negative relationship exists. The extent to which the dots in a scatterplot cluster together in the form of a line indicates the strength of the relationship. Scatterplots with dots that are spread apart represent a weak relationship. Below are scatterplots for arm and grip strength against supervisor ratings and work simulations. Interpretation: Which of the following statements can we conclude is NOT true? The scatterplots above indicate that strength scores (arm and grip) tend to be positively related with ratings and work simulation.

Individuals with lower strength scores tended to receive lower ratings and perform worse on the simulations than stronger individuals. The scatterplots indicate that arm and grip scores were more strongly related to work simulation scores than supervisory ratings. Grip scores and work simulation scores appear to have a lower correlation coefficient than grip scores and ratings. Correlations Although scatterplots give you a general feel for the extent to which a relationship exists between two variables, they leave a lot of room for interpretation.

For this reason one usually computes a correlation coefficient to determine the degree of linear relationship between two variables. Correlations coefficients range from -1 to 1. Correlations closer to zero indicate weak relationships whereas those closer to 1 and -1 indicate a strong positive and negative relationship, respectively. Below is a correlation matrix summarizing the various correlations among the study variables. From the table, you can see that all correlations are positive indicating that higher scores in one variable are always associated with higher scores on the other.

The strongest correlation observed was between arm strength and work simulations (r = . 686). The weakest relationship was between ratings and work simulations (r = . 1681). Now examine the rest of the correlations. – – Correlation Coefficients – – ARM GRIP RATINGS SIMS ARM 1. 0000 . 6298 . 2213 . 6860 ( 147) ( 147) ( 147) ( 147) P= . P= . 000 P= . 007 P= . 000 GRIP . 6298 1. 0000 . 1833 . 6398 ( 147) ( 147) ( 147) ( 147) P= . 000 P= . P= . 026 P= . 000 RATINGS . 213 . 1833 1. 0000 . 1681 ( 147) ( 147) ( 147) ( 147) P= . 007 P= . 026 P= . P= . 042 SIMS . 6860 . 6398 . 1681 1. 0000 ( 147) ( 147) ( 147) ( 147) P= . 000 P= . 000 P= . 042 P= . (Coefficient / (Cases) / 2-tailed Significance) Regression Another concept related to correlation is linear regression. This procedure is used to derive the actual equation of the best fitting line through the points on a scatterplot. Regression also allows you to determine how well one variable can be used to predict another.

Below are SPSS outputs for each strength test (arm and strength) predicting each of the two performance measures (ratings and simulations). There are several numbers that are particularly noteworthy. First, the R-Square indicates the proportion of variance in the dependent variable explained by the independent variable. Thus, for predicting Ratings from Arm strength, you can see that the linear equation predicts . 048 or approximately 5% of the variance in ratings. Next, the Standard Error indicates how far off you would be, on average, if you were to use the independent variable to predict scores on the dependent variable.

Thus, if you used Arm strength scores you could predict ratings with an average error of 8. 34 (on a 60-point scale). The specific equation for the line of best fit can be derived from the numbers under the “B” column. The first number indicates the slope of the line (. 089 for the first example) and the second number indicates the intercept (33. 97 for the first example). Thus, one could get a predicted Ratings score by plugging in a person’s Arm score into the equation: Ratings = . 089*Arm + 33. 97 The regression outputs for the other strength scores and performance measures are presented below. Regression Equation of ARM predicting RATINGS

Multiple R . 22128 R Square . 04896 Adjusted R Square . 04241 Standard Error 8. 33922 Analysis of Variance DF Sum of Squares Mean Square Regression 1 519. 16642 519. 16642 Residual 145 10083. 67246 69. 54257 F = 7. 46545 Signif F = . 0071 —————— Variables in the Equation —————— Variable B SE B Beta T Sig T ARM . 089331 . 032694 . 221280 2. 732 . 0071 (Constant) 33. 974907 2. 665032 12. 748 . 0000

What is the predicted SIMS score given an ARM score of 110? 1. 659752 3 2. 02538 1. 90677 Regression Equation of ARM predicting SIMS Multiple R . 68601 R Square . 47061 Adjusted R Square . 46696 Standard Error 1. 22582 Analysis of Variance DF Sum of Squares Mean Square Regression 1 193. 68606 193. 68606 Residual 145 217. 88128 1. 50263 F = 128. 89808 Signif F = . 0000 —————— Variables in the Equation —————— Variable B SE B Beta T Sig T ARM . 054563 . 04806 . 686007 11. 353 . 0000 (Constant) -4. 095160 . 391745 -10. 454 . 0000 Regression Equation of GRIP predicting RATINGS Multiple R . 18326 R Square . 03358 Adjusted R Square . 02692 Standard Error 8. 40639 Analysis of Variance DF Sum of Squares Mean Square Regression 1 356. 07735 356. 07735 Residual 145 10246. 76153 70. 66732 F = 5. 03878 Signif F = . 0263 —————— Variables in the Equation —————— Variable B SE B Beta T Sig T

GRIP . 066090 . 029442 . 183257 2. 245 . 0263 (Constant) 33. 724714 3. 318697 10. 162 . 0000 What is the proportion of varance in the Ratings variable explained by the Grip variable? .03358 .1826 8. 04639 .029442 Regression Equation of GRIP predicting SIMS Multiple R . 63985 R Square . 40940 Adjusted R Square . 40533 Standard Error 1. 29474 Analysis of Variance DF Sum of Squares Mean Square Regression 1 168. 49674 168. 49674 Residual 145 243. 07060 1. 67635 F = 100. 51412 Signif F = . 000 —————— Variables in the Equation —————— Variable B SE B Beta T Sig T GRIP . 045463 . 004535 . 639846 10. 026 . 0000 (Constant) -4. 809675 . 511141 -9. 410 . 0000 If we use the Grip variable to predict the SIMS variable, how far, on average, would the predicted value be from the actual value? .63985 .40940 1. 29474 .045463 RAW DATA GRIPARMRATINGSSIMS 105. 580. 531. 81. 18 106. 59339. 80. 94 948146. 80. 84 90. 533. 552. 2-2. 45 10447. 531. 21 171125. 546. 64. 38 107. 581. 529. 8-0. 38 124. 583. 539-0. 01 876550. 6-0. 99 102. 77. 540. 1-0. 04 11954. 529. 6-1. 15 5468. 552. 60. 19 11910151. 80. 58 88. 5109. 5351. 2 157. 5103. 545. 81. 22 11079. 532. 4-0. 15 92. 58538. 151. 26 99. 591. 5290. 92 10963. 542-0. 09 1149451-0. 67 9296. 538. 6-1. 17 13884. 5491. 84 91. 56428-0. 62 18912249. 2664. 87 115101. 545. 60. 77 1218452. 62. 97 128. 593. 546. 43. 16 12884460. 27 823931. 3-3. 88 128. 588570. 91 11870. 526. 40. 91 104. 569. 538. 2-0. 94 122. 566. 530. 2-2. 17 882533. 8-2. 05 10563. 526. 8-1. 3 127104. 541. 60. 26 120. 58632. 40. 13 10385. 553. 80. 71 115. 53647. 2-1. 08 7869. 532. 4-1. 67 94. 588. 545. 60. 72 996236. 334-0. 61 11596. 554. 2-1. 11 12290. 39. 80. 16 91. 57142. 5-0. 79 117. 561. 5440. 5 998052. 40. 49 121. 5106. 5422. 3 99. 54441. 3-2. 34 897738. 80. 51 92. 58545-0. 42 124. 510642. 80. 79 130. 5104. 531. 41. 83 10352. 538. 4-0. 8 10389. 537. 4-1. 21 153132425. 17 111. 59550. 80. 8 13177. 547-1. 12 12958. 534. 80. 85 1128325. 40. 2 9497. 531. 60. 44 10694. 527. 2-0. 15 109. 5104. 545-2. 04 118. 594. 552. 4-0. 01 120. 59548. 43. 09 772943. 6-3. 38 1308450. 41. 85 118. 5113. 554. 62. 45 98. 569. 521. 6-0. 3 9361240. 76 764750. 60 80. 53622. 6-2. 65 858351. 4-1. 14 114. 582. 547. 2-1. 16 12879. 529. 20. 38 14610847. 62. 18 12510837. 21. 01 1536739. 2-0. 05 10364. 40. 8-0. 41 81. 55524-0. 57 10770. 548. 4-0. 68 1139650. 40. 42 123. 56942. 3-0. 36 827855-1. 65 100. 55130. 20. 77 9564. 536-1. 97 75. 557. 542. 10. 35 1319350. 21. 92 98. 565. 529. 6-1. 29 119. 588. 532. 60. 47 9975. 540. 8-0. 38 1018942. 21. 35 152. 597452. 6 6448. 545. 2-0. 55 9159. 537-1. 47 12396. 532. 81. 47 111. 570. 555. 6-0. 29 112. 569. 544. 42. 37 86. 56035. 8-0. 4 14211526. 62. 69 7166380. 2 95. 58941. 4-1. 1 136. 59036. 52. 13 94. 562. 556. 6-1. 59 90. 564. 547. 8-2. 79 111. 595. 5331. 53 11110151. 6020. 94 11982. 531. 41. 21 119. 590. 548. 43. 04 134103. 5463. 51 1348948. 41. 62 132. 583. 537-0. 62 121. 58945. -0. 89 91. 57333. 6-0. 82 173. 511754. 20. 68 128. 580561. 43 144. 59435. 21. 57 291928-4. 17 135101. 538. 44. 52 896049. 4-1. 92 87101. 540-1. 51 104. 56938-1. 43 12010052. 41. 24 117. 556. 534. 8-0. 57 10075. 532. 40. 03 925739. 5-1. 64 124. 58244. 6-0. 29 103. 57748. 8-0. 5 54. 53539. 8-1. 68 13690. 543. 61. 11 124. 576. 533. 2-1. 11 97. 558. 552. 8-0. 35 1146234. 4-1. 87 127. 595. 537. 40. 73 10468. 535. 6-0. 92 14111335. 54. 95 84. 58330. 6-1. 09 12366. 548. 80. 32 133. 587. 5400. 21 132. 59947. 41. 64 82. 557. 544. 8-0. 25 13186371. 03 1477157. 20. 53 109. 586. 5432. 89 5467. 541. 7-1. 38 12663. 5371. 33 943837. 3-1. 53

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