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Ch. 10 Correlation and Regression 10-1 Notes Scatter Diagrams and Linear Correlation

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Scatter Diagram – a graph in which data pairs (x, y) are plotted as individual points on a grid with ________________ axis x and ____________ axis y. We call x the ______________ variable and y the ______________ variable. Line of “best fit” – a line that runs __________ the data points and is ___________, on average, to the data points. Will cover in more detail in 10-2.

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Ex. 1 A large industrial plant has seven divisions that do the same type of work. A safety inspector visits each division of 20 workers quarterly. The number x of work-hours devoted to safety training and the number y of work-hours lost due to industry-related accidents are recorded for each separate division below: Safety Report Division1234567 x10.019.530.045.050.065.080.0 y80656855351012

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a) Make a scatter diagram for these pairs. b) As the number of hours spent training increases, what happens to the number of hours lost to accidents?

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c) Does a line fit the data reasonably well? d) Draw a line that you think “fits best”.

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Correlation – describes/quantifies how well the data _____ to the ____________________. Ex. 2Using the diagrams below answer the following: I.II.III. a) Which has no linear correlation? b) Which has perfect linear correlation? c) Which can reasonably be fitted by a straight line?

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Correlation Positive Correlation – low values of x are associated with _____ values of y and high values of x are associated with _____ values of y. Negative Correlation – low values of x are associated with _____ values of y and high values of x are associated with _____ values of y.

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Which of the above are negatively correlated? Which of the above are positively correlated?

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Sample Correlation Coefficient (r) – a mathematical measurement that describes the _____________ of the __________ association between two variables of a sample. 1.r is a unitless measurement between ___ and ___. In symbols, ___ < r < ___. If r = ___, there is perfect positive linear correlation. If r = ___, there is perfect negative linear correlation. If r = ___, there is no linear correlation. The closer r is to ___ or ___, the better a line describes the relationship between the two variables x and y.

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2.Positive values of r imply that as x increases, y tends to ____________. Negative values of r imply that as x increases, y tends to ____________. 3.The value of r is the same regardless of which variable is the _____________ variable and which is the ___________ variable. 4.the value of r does not change when either variable is ____________________________.

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Tech Notes To find r using TI 83/84 First use CATALOG, find Diagnostic On, and press Enter twice. Then, when you use STAT, CALC, option 8:LinReg(a+bx), the value of r will be given.

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Ex. 1 In one of the Boston city parks there has been a problem with muggings in the summer months. A police cadet took a random sample of 10 days (out of the 90-day summer) and compiled the following data. For each day, x represents the number of police officers on duty in the park and y represents the number of reported muggings on that day. X (#of police on duty)1015161461812147 Y (#of reported muggings)5219781536 a) Construct a scatter diagram b) From scatter diagram, will r be positive, negative, or zero? Explain. c) Find correlation coefficient (r).

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Final thoughts about correlation 1. r = ________ correlation coefficient; whereas ρ = __________ correlation coefficient. 2. Just because two variables have a strong correlation does not imply that one ________ the other. Sometimes the strong relationship can be caused by another variable (________ ______________). 3. Correlation between two variables consisting of ____________ is usually stronger than between _____________ values.

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Assignment Day 1 P. 503 #1, 2, 3, 5, 7, 9, 11, 13, 14 Day 2 P. 504 #6, 8, 10, 12, 15, 16

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