ST 361 Normal Distribution
ST361: Ch3 Bivariate Data Analysis
Overview/Review of Bivariate Data Analysis
• Bivariate data: 2 variables X and Y involved. Usually denote
← X as ___________________ variable, aka _______________ variable
← Y as ___________________ variable, aka _______________ variable
• Bivariate data analysis: interested in the relationship between X and Y
• Overview of bivariate data analysis:
| |Example 1 |Example 2 |Example 3 |
| |Obs |Obs |Obs |
| |Battery Brand |Study Time (hr) |Drug Type |
| |Lifetime (hr) |Exam Score |Side Effect |
| | | | |
| |1 |1 |1 |
| |Duracell |6.5 |New |
| |4.2 |76 |Y |
| | | | |
| |2 |2 |2 |
| |Eveready |7.3 |New |
| |5.1 |83 |Y |
| | | | |
| |3 |3 |3 |
| |Eveready |9.5 |New |
| |3.9 |92 |N |
| | | | |
| |[pic] |4 |[pic] |
| | |7.1 | |
| | |87 | |
| | | | |
| |N |5 |N |
| |Duracell |8.4 |Old |
| |3.8 |93 |N |
| | | | |
| | |6 | |
| | |8.0 | |
| | |88 | |
| | | | |
|Question of interest |Do the two brands have the same |Is exam score related to study time? |Can new drug reduce side effect? |
| |lifetime? | | |
|Independent variable X | | | |
|Dependent variable Y | | | |
|Type of Variables |X: |X: |X: categorical |
| |Y: |Y: |Y: categorical |
|Graphical Presentation | Side-by-side Boxplot |Scatter plot | Bar plot |
| |[pic] |[pic] |[pic] |
| | | | |
| | | | |
| | | | |
| | | | |
|Numerical Summary | | | |
| | | | |
| | | | |
| | | | |
| | | | |
|Statistical Inference |[pic] |Population regression line |[pic] |
| | |[pic] | |
| | |[pic] | |
| | |Population Correlation coefficient | |
| | |[pic] | |
| | |[pic] | |
------------------------------------------------------------------------------------------------------------------
ST361: Ch3.2 Correlation Coefficient
Topics:
a) Definition
b) Interpretation
c) Calculation
------------------------------------------------------------------------------------------------------------------
a) Definition: The sample correlation coefficient r is a statistic that quantifies the ______________ and ______________ of the _____________________________ between 2 continuous variables X and Y
• The ____________ of r indicates the strength of the relationship:
The correlation coefficient r takes values in the range of _____________________
• The ____________ of r indicates the direction of the relationship between X and Y:
Ex.
[pic]
b) Interpretation:
(1) 0.8 < | r | < 1 : _______________ relationship between X and Y
(2) 0.5 < | r | [pic] 0.8 : _______________ relationship
(3) 0.0 < | r | [pic] 0.5 : _______________ relationship
Comments:
• The value of r ____________________________________________________________
• Meaning of r = 0:
[pic]
c) Calculation
Q: if the definitions of X and Y are swapped, will the value of r change?
|Ex. Study time vs. Exam score |[pic] |
| | |
|X | |
|Y | |
|[pic] | |
| | |
|Obs | |
|Study Time (hr) | |
|Exam Score | |
| | |
| | |
|1 | |
|6.5 | |
|76 | |
|494 | |
| | |
|2 | |
|7.3 | |
|83 | |
|605.9 | |
| | |
|3 | |
|9.5 | |
|92 | |
|874 | |
| | |
|4 | |
|7.1 | |
|87 | |
|617.7 | |
| | |
|5 | |
|8.4 | |
|93 | |
|781.2 | |
| | |
|6 | |
|8.0 | |
|88 | |
|704.0 | |
| | |
| | |
| | |
|[pic] | |
|[pic] | |
|[pic]=4076.8 | |
|Calculate the sample correlation coefficient r. | |
Ex. Body Mass Index vs. Blood Pressure
| |X |Y |[pic] |
|Obs |Body Mass Index |Systolic Blood Pressure | |
|1 |18 |120 |2160 |
|2 |20 |110 |2200 |
|3 |22 |120 |2640 |
|4 |25 |135 |3375 |
|5 |26 |140 |3640 |
|6 |29 |115 |3335 |
|7 |30 |150 |4500 |
|8 |33 |165 |5445 |
|9 |33 |160 |5280 |
|10 |35 |180 |6300 |
[pic],[pic], [pic]=38875
Calculate the sample correlation coefficient r.
[pic]
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