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2 Variable Stats Does the amount of time spent on homework improve your grade? If you eat more junk food will you gain weight? Does the amount of rainfall affect the amount of wheat yielded each year?

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By using a scatter plot we can view a visual pattern in the graph By using a scatter plot we can view a visual pattern in the graph This pattern can reveal the nature of the relationship between the two variables This pattern can reveal the nature of the relationship between the two variables Time ( months ) Weight (kg) Weight Change after Exercise

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Variables When Graphing this information you must determine which variable goes on which axis When Graphing this information you must determine which variable goes on which axis One Variable is the dependent (or response) variable One Variable is the dependent (or response) variable The other is the independent (or explanatory) variable The other is the independent (or explanatory) variable

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Which Variable goes Where? x y Independent Dependent

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Independent or Dependent Variable? For each relationship, sketch the axis and label the independent and dependent variables a) The number of hours a student studies per course and his or her average mark in the course b) The time of day and the number of cups of coffee sold c) The amount of snowfall and the number of collisions on major highways

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Correlation Variables have a LINEAR CORRELATION if changes in one variable tend to be proportional to the changes in another variable Variables have a LINEAR CORRELATION if changes in one variable tend to be proportional to the changes in another variable If your Y increases at a constant rate as X increases then you have a Perfect Positive (or direct) linear correlation If your Y increases at a constant rate as X increases then you have a Perfect Positive (or direct) linear correlation

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If the Y decreases at a constant rate as X increases then you have a Perfect Negative (or inverse) linear correlation If the Y decreases at a constant rate as X increases then you have a Perfect Negative (or inverse) linear correlation

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Strength of Correlation The stronger the correlation the more closely the data points cluster around the line of best fit. The stronger the correlation the more closely the data points cluster around the line of best fit. State the strength *strong, weak, moderate, moderately strong etc….*State the strength *strong, weak, moderate, moderately strong etc….* State direction *positive or negative*State direction *positive or negative* State type of correlation *Linear, Non- linear, exponential etc…*State type of correlation *Linear, Non- linear, exponential etc…*

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Turn to page 160 in your textbook Turn to page 160 in your textbook Example 1Classifying Linear Relationships Example 1Classifying Linear Relationships

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Correlation Coefficient Karl Pearson (1857-1936) developed a formula to determine a value that would tell us the strength of a correlation. Karl Pearson (1857-1936) developed a formula to determine a value that would tell us the strength of a correlation. Note he also developed the standard deviation Note he also developed the standard deviation

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Correlation Covariance First need to define what we call the covariance of two variables in a sample. First need to define what we call the covariance of two variables in a sample.

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Correlation Coefficient Denoted with an r Denoted with an r Is the covariance divided by the product of the standard deviations for X and Y Is the covariance divided by the product of the standard deviations for X and Y

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Gives a quantitative measure of the strength of a relationship Gives a quantitative measure of the strength of a relationship Basically shows how close the data points cluster around the line of best fit. Basically shows how close the data points cluster around the line of best fit. Also called the Pearson product- moment coefficient of correlation (PPMC) or Pearson’s r Also called the Pearson product- moment coefficient of correlation (PPMC) or Pearson’s r

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Strength of Correlation The closer the correlation coefficient (r ) is to 1 or -1 the stronger the correlation The closer the correlation coefficient (r ) is to 1 or -1 the stronger the correlation r = -1 r =+1

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No relation has a correlation that is close to zero and the dots are scattered everywhere No relation has a correlation that is close to zero and the dots are scattered everywhere r = 0

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The following diagram illustrates how the correlation coefficient corresponds to the strength of a linear correlation The following diagram illustrates how the correlation coefficient corresponds to the strength of a linear correlation -1 -0.67 -0.3300.33 0.671 Positive Linear CorrelationNegative Linear Correlation WeakModerateStrong Perfect WeakModerateStrong Perfect

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SO LONG!!!!!!!! There is a quicker way to calculate r, without having to find the standard deviations……but it looks more difficult There is a quicker way to calculate r, without having to find the standard deviations……but it looks more difficult

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Example A Farmer wants to determine whether there is a relationship between the mean temperature during the growing season and the size of his wheat crop. He assembles the following data for the last six crops Mean Temp ( °C) Yield (tonnes/hectare) 41.6 82.4 102.0 92.6 112.1 62.2

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a) Does the Scatter plot of this data indicate any linear correlation between the two variables 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5Mean Temperature ( ºC) Yield (T/ha) Yield of Growing Season

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b) Compute the Correlation Coefficient TempYield X2X2X2X2 Y2Y2Y2Y2xy 41.6 82.4 102.0 92.6 112.1 62.2

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c) What can the farmer conclude about the relationship between the mean temperatures during the growing season and the wheat yields on his farm

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Homework Pg 168 #1-6

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