Regression & correlation
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This document provides an overview of key concepts in business statistics including correlation, scatter plots, the correlation coefficient, linear regression, and calculating the regression line. It defines correlation as a linear association between two variables and explains how scatter plots can show positive, negative, or no relationship. The correlation coefficient r measures the strength and direction of a relationship between -1 and 1. Linear regression finds the linear relationship between a dependent and independent variable to predict future values using the regression line equation y=a+bx.
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This document introduces correlation and regression. It defines correlation as a measure of the association between two numerical variables and discusses how the Pearson correlation coefficient measures both the direction and strength of the linear relationship between two variables. Regression is introduced as a statistical method for finding the line of best fit for one variable based on another. Simple linear regression finds the line of best fit for one dependent variable based on one independent variable. The document provides steps for performing simple linear regression using a TI-83 graphing calculator.
Properties of Standard Deviation
The document discusses standard deviation and its properties. Standard deviation is a measure of how spread out numbers are from the average (mean) value. It is always non-negative and can be impacted by outliers. A low standard deviation means values are close to the mean, while a high standard deviation means values are more spread out. Standard deviation can be used to calculate what percentage of data falls within certain intervals from the mean when data is normally distributed.
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This document provides an overview of the banking system in Nepal. It begins by explaining the purpose of banks and then outlines the different types of banks in Nepal, including central banks, commercial banks, development banks, finance companies, and microcredit development banks. A total of 30 commercial banks, 82 development banks, 48 finance companies, and 37 microcredit development banks currently operate in Nepal. The document also includes organizational charts of the banking hierarchy and describes some of the key roles and services provided by banks, such as accepting deposits, lending money, remittances, safe deposit services, and capital market activities.
Chapter 11
The document discusses different types of two-sample hypothesis tests, including tests comparing two population means of independent samples, two population proportions, and paired or dependent samples. It provides examples and step-by-step explanations of how to conduct two-sample t-tests, z-tests, and tests of proportions. Key points covered include determining the appropriate test statistic based on sample size and characteristics, stating the null and alternative hypotheses, test criteria, and decisions rules.
Chapter 08
1. The document discusses sampling methods and the central limit theorem. It describes various probability sampling methods like simple random sampling, systematic random sampling, and stratified random sampling.
2. It defines the sampling distribution of the sample mean and explains that according to the central limit theorem, the sampling distribution will follow a normal distribution as long as the sample size is large.
3. The mean of the sampling distribution is equal to the population mean, and its variance is equal to the population variance divided by the sample size. This allows probabilities to be determined about a sample mean falling within a certain range.
Chapter 11 ,Measures of Dispersion(statistics)
This document discusses various measures of dispersion, which describe how spread out or varied the values in a data set are. It describes absolute measures like range and relative measures like coefficient of range. It also discusses interquartile range, mean deviation, standard deviation, and the Lorenz curve. Quartile deviation and mean deviation are simple measures but are less accurate and reliable. Standard deviation is a more certain measure but gives more importance to extreme values. The Lorenz curve measures deviation from equal distribution for variables like income, wealth, wages and more.
Chapter 12
This document discusses analysis of variance (ANOVA) techniques. It defines the F-distribution and its characteristics. It then covers testing for equal variances between two populations and comparing means of two or more populations using one-way and two-way ANOVA. Examples are provided to illustrate hypothesis testing using the F-statistic to compare variances and population means. Finally, it discusses developing confidence intervals for differences in treatment means and using ANOVA in Excel.
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Regression & correlation
- 1. Course Title: Business Statistics BBA (Hons) 2nd Semester Course Instructor: Atiq ur Rehman Shah Lecturer, Federal Urdu University of Arts, Science & Technology, Islamabad +92-345-5271959 aatresh@gmail.com
- 2. Correlation • Correlation is a LINEAR association between two random variables • Correlation is a statistical technique used to determine the degree to which two variables are related
- 3. Scatter diagram • Rectangular coordinate • Two quantitative variables • One variable is called independent (X) and the second is called dependent (Y)
- 5. Scatter diagram of weight and systolic blood pressure
- 6. Scatter diagram of weight and systolic blood pressure
- 7. Scatter plots The pattern of data is indicative of the type of relationship between your two variables: • positive relationship • negative relationship • no relationship
- 10. No relation
- 11. Correlation Coefficient • The correlation coefficient (r) measures the strength and direction of relationship between two variables
- 12. How to interpret the value of r? • r lies between -1 and 1. Values near 0 means no (linear) correlation and values near ± 1 means very strong correlation. • The negative sign means that the two variables are inversely related, that is, as one variable increases the other variable decreases.
- 13. How to interpret the value of r?
- 14. Pearson’s r • A 0.9 is a strong positive association (as one variable rises, so does the other) • A -0.9 is a strong negative association (as one variable rises, the other falls) r=correlation coefficient
- 15. Coefficient of Determination Defined • Pearson’s r can be squared , r 2, to derive a coefficient of determination. • Coefficient of determination – the portion of variability in one of the variables that can be accounted for by variability in the second variable
- 16. • Example of depression and CGPA – Pearson’s r shows negative correlation, r=-0.5 – r2=0.25 • In this example we can say that 1/4 or 0.25 of the variability in CGPA scores can be accounted for by depression (remaining 75% of variability is other factors, habits, ability, motivation, courses studied, etc)
- 17. Coefficient of Determination and Pearson’s r • If r=0.5, then r2=0.25 • If r=0.7 then r2=0.49 • Thus while r=0.5 versus 0.7 might not look so different in terms of strength, r2 tells us that r=0.7 accounts for about twice the variability relative to r=0.5
- 18. Example • Calculate the coefficient of correlation between the value X and Y given below: X 78 89 97 69 59 79 68 61 Y 125 137 156 112 107 136 123 108
- 19. X Y X2 Y2 XY 78 125 6084 15625 9750 89 137 7921 18769 12193 97 156 9409 24336 15132 69 112 4761 12544 7728 59 107 3481 11449 6313 79 136 6241 18496 10744 68 123 4624 15129 8364 61 108 3721 11664 6588 Summation 600 1004 46242 128012 76812
- 20. = 0.95 Hence the correlation co-efficient between X and Y is 0.95. ** (What does this value tells us??)**
- 21. Regression • A statistical tool that is used to investigate the dependence of one variable (dependent variable) on one or more other variables (independent variables) • The dependent variable (Y) is the variable for which we want to make a prediction. • The independent variable (X) is the variable on the basis of which we are making predictions.
- 22. • The linear relationship between two variables can either be positive or negative. • For instance, an increase in advertisement budget will bring more sales (positive), and increase in temperature will decrease the cooling efficiency of a room AC (negative)
- 23. Simple Linear Regression • Positive Linear RelationshipPositive Linear Relationship yy xx Slope (b)Slope (b) is positiveis positive Regression lineRegression line InterceptIntercept (a)(a)
- 24. Simple Linear Regression • Negative Linear RelationshipNegative Linear Relationship yy xx Slope (b)Slope (b) is negativeis negative Regression lineRegression line InterceptIntercept (a)(a)
- 25. Simple Linear Regression • No RelationshipNo Relationship yy xx Slope (b)Slope (b) is 0is 0 Regression lineRegression line InterceptIntercept (a)(a)
- 26. Simple Linear Regression Equation • Hence the equation for linear regression line can be written as: y= a + bx Where: y= dependent variable x= independent variable a= y-intercept (i.e value of y when x=0) b= slope
- 27. Least-squares estimates • For a simple linear regression equation: y= a + bx We have, Where, and
- 28. Example • Compute the least squares regression equation of Y on X for the following data. What is the regression coefficient and what does it mean?? X 5 6 8 10 12 13 15 16 17 Y 16 19 23 28 36 41 44 45 50
- 29. X Y XY X2 5 16 80 25 6 19 114 36 8 23 184 64 10 28 280 100 12 36 432 144 13 41 533 169 15 44 660 225 16 45 720 256 17 50 850 289 Summation 102 302 3853 1308
- 30. Now = 102/9 = 11.33 And = 302/9 = 33.56 = 9(3853) – (102) (302) 9( 1308) – (102)2 = 3873/1368 So b = 2.381
- 31. And = 33.56 – (2.831) (11.33) = 1.47 Hence the desired estimated regression line of Y on X is y= 1.47 + 2.831x ** The estimated regression co-efficient is b=2.831, which means that yhe value of y increase by 2.831 units for a unit increase in x.