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Graphing Data 4
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IntroductionLecture1.1
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Basic ChartsLecture1.2
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Basic Charts Part 2Lecture1.3
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RandomnessLecture1.4
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Mean and Standard Deviation 5
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IntroductionLecture2.1
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Random WalkLecture2.2
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Random Walk Part 2Lecture2.3
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Mean and Standard DeviationLecture2.4
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Mean and Standard Deviation Part 2Lecture2.5
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Distributions 6
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IntroductionLecture3.1
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PDF and CDFLecture3.2
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PDF and CDF Part 2Lecture3.3
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Z ScoresLecture3.4
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PlottingLecture3.5
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SolutionLecture3.6
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Correlation and Linear Regression 7
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IntroductionLecture4.1
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BasicsLecture4.2
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CorrelationLecture4.3
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Z-ScoreLecture4.4
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Z-Score Part 2Lecture4.5
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Linear RegressionLecture4.6
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Linear Regression Part 2Lecture4.7
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Probability 3
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IntroductionLecture5.1
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Probability BasicsLecture5.2
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ProbabilityLecture5.3
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Counting Principles 3
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IntroductionLecture6.1
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Factorials and PermutationsLecture6.2
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Binomial TheoremLecture6.3
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Binomial Distribution 3
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IntroductionLecture7.1
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Binomial DistributionLecture7.2
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Binomial Distribution Part 2Lecture7.3
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Confidence Interval 7
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IntroductionLecture8.1
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AlphaLecture8.2
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Confidence IntervalsLecture8.3
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Confidence Intervals Part 2Lecture8.4
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Confidence Intervals Part 3Lecture8.5
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Confidence Intervals Part 4Lecture8.6
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Confidence Intervals Part 5Lecture8.7
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Proportion Confidence Interval 3
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IntroductionLecture9.1
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EquationsLecture9.2
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SolutionLecture9.3
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Hypothesis Testing 5
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IntroductionLecture10.1
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The Null HypothesisLecture10.2
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The Null Hypothesis Part 2Lecture10.3
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One Tail TestLecture10.4
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One Tail Test Part 2Lecture10.5
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Comparing Two Means 5
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IntroductionLecture11.1
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Known Standard DeviationsLecture11.2
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Unknown Standard DeviationsLecture11.3
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Same Unknown Standard DeviationsLecture11.4
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SolutionLecture11.5
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Chi-squared Test 3
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IntroductionLecture12.1
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BasicsLecture12.2
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Expected ValuesLecture12.3
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Z-Score Part 2
Solution
Solution
Starting Equation
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y}}{\sqrt{\Sigma (x-\overline{x})^2 \Sigma (y-\overline{y})^2}}$$
Step 1
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{\sqrt{ \Sigma (x-\overline{x})^2} \sqrt{ \Sigma (y-\overline{y})^2}}$$
Step 2 (Look at Variance Equation)
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{\sqrt{ n\sigma_{y}^2} \sqrt{ n\sigma_{y}^2}}$$
Step 3
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{n\sigma_{y} \sigma_{y}}$$
Step 4
$$r_{xy} = \frac{\Sigma z_{x}z_{y}}{n} $$
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y}}{\sqrt{\Sigma (x-\overline{x})^2 \Sigma (y-\overline{y})^2}}$$
Step 1
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{\sqrt{ \Sigma (x-\overline{x})^2} \sqrt{ \Sigma (y-\overline{y})^2}}$$
Step 2 (Look at Variance Equation)
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{\sqrt{ n\sigma_{y}^2} \sqrt{ n\sigma_{y}^2}}$$
Step 3
$$r_{xy} = \frac{\Sigma (x-\overline{x}) (y-\overline{y})}{n\sigma_{y} \sigma_{y}}$$
Step 4
$$r_{xy} = \frac{\Sigma z_{x}z_{y}}{n} $$
Let’s check with programming.
z1 = (xVals-xVals.mean())/xVals.std()
z2 = (yVals-yVals.mean())/yVals.std()
print(sum(z1*z2)/(100))
print(np.corrcoef(xVals,yVals)[0][1])At this time, we will also explain what covariance is, and how is relates to the above equation.
Equation
$$ Cov_{x,y} = \frac{\Sigma (x_i-\overline{x})(y_i-\overline{y})}{n} $$
This equation describes the way in which the variances of two variables move together, but not in a normalized way. You’ll notice that the normalized way is our correlation which divides covariance by the standard deviation of both variables.
Equation
$$ r_{x,y} = \frac{Cov_{x,y} }{\sigma_{x}\sigma_{y}}$$
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