statistics pooled variance r


Once the degree of relationship between variables has been established using co-relation analysis, it is natural to delve into the nature of relationship. Regression analysis helps in determining the cause and effect relationship between variables. It is possible to predict the value of other variables (called dependent variable) if the values of independent variables can be predicted using a graphical method or the algebraic method.

Graphical Method

It involves drawing a scatter diagram with independent variable on X-axis and dependent variable on Y-axis. After that a line is drawn in such a manner that it passes through most of the distribution, with remaining points distributed almost evenly on either side of the line.

A regression line is known as the line of best fit that summarizes the general movement of data. It shows the best mean values of one variable corresponding to mean values of the other. The regression line is based on the criteria that it is a straight line that minimizes the sum of squared deviations between the predicted and observed values of the dependent variable.

Algebraic Method

Algebraic method develops two regression equations of X on Y, and Y on X.

Regression equation of Y on X

${Y = a+bX}$

Where −

  • ${Y}$ = Dependent variable
  • ${X}$ = Independent variable
  • ${a}$ = Constant showing Y-intercept
  • ${b}$ = Constant showing slope of line

Values of a and b is obtained by the following normal equations:

${\sum Y = Na + b\sum X \\[7pt]
\sum XY = a \sum X + b \sum X^2
}$

Where −

  • ${N}$ = Number of observations

Regression equation of X on Y

${X = a+bY}$

Where −

  • ${X}$ = Dependent variable
  • ${Y}$ = Independent variable
  • ${a}$ = Constant showing Y-intercept
  • ${b}$ = Constant showing slope of line

Values of a and b is obtained by the following normal equations:

${\sum X = Na + b\sum Y \\[7pt]
\sum XY = a \sum Y + b \sum Y^2
}$

Where −

  • ${N}$ = Number of observations

Example

Problem Statement:

A researcher has found that there is a co-relation between the weight tendencies of father and son. He is now interested in developing regression equation on two variables from the given data:

Weight of father (in Kg) 69 63 66 64 67 64 70 66 68 67 65 71
Weight of Son (in Kg) 70 65 68 65 69 66 68 65 71 67 64 72

Develop

  1. Regression equation of Y on X.
  2. Regression equation of on Y.

Solution:

${X}$ ${X^2}$ ${Y}$ ${Y^2}$ ${XY}$
69 4761 70 4900 4830
63 3969 65 4225 4095
66 4356 68 4624 4488
64 4096 65 4225 4160
67 4489 69 4761 4623
64 4096 66 4356 4224
70 4900 68 4624 4760
66 4356 65 4225 4290
68 4624 71 5041 4828
67 4489 67 4489 4489
65 4225 64 4096 4160
71 5041 72 5184 5112
${\sum X = 800}$ ${\sum X^2 = 53,402}$ ${\sum Y = 810}$ ${\sum Y^2 = 54,750}$ ${\sum XY = 54,059}$

Regression equation of Y on X

Y = a+bX

Where , a and b are obtained by normal equations

${\sum Y = Na + b\sum X \\[7pt]
\sum XY = a \sum X + b \sum X^2 \\[7pt]
Where\ \sum Y = 810, \sum X = 800, \sum X^2 = 53,402 \\[7pt]
, \sum XY = 54, 049, N = 12 }$

${\Rightarrow}$ 810 = 12a + 800b … (i)

${\Rightarrow}$ 54049 = 800a + 53402 b … (ii)

Multiplying equation (i) with 800 and equation (ii) with 12, we get:

96000 a + 640000 b = 648000 … (iii)

96000 a + 640824 b = 648588 … (iv)

Subtracting equation (iv) from (iii)

-824 b = -588

${\Rightarrow}$ b = -.0713

Substituting the value of b in eq. (i)

810 = 12a + 800 (-0.713)

810 = 12a + 570.4

12a = 239.6

${\Rightarrow}$ a = 19.96

Hence the equation Y on X can be written as

${Y = 19.96 – 0.713X}$

Regression equation of Y on X

X = a+bY

Where , a and b are obtained by normal equations

${\sum X = Na + b\sum Y \\[7pt]
\sum XY = a \sum Y + b \sum Y^2 \\[7pt]
Where\ \sum Y = 810, \sum Y^2 = 54,750 \\[7pt]
, \sum XY = 54, 049, N = 12 }$

${\Rightarrow}$ 800 = 12a + 810a + 810b … (V)

${\Rightarrow}$ 54,049 = 810a + 54, 750 … (vi)

Multiplying eq (v) by 810 and eq (vi) by 12, we get

9720 a + 656100 b = 648000 … (vii)

9720 a + 65700 b = 648588 … (viii)

Subtracting eq viii from eq vii

900b = -588

${\Rightarrow}$ b = 0.653

Substituting the value of b in equation (v)

800 = 12a + 810 (0.653)

12a = 271.07

${\Rightarrow}$ a = 22.58

Hence regression equation of X and Y is

${X = 22.58 + 0.653Y}$

Table of Contents
1.statistics adjusted rsquared

2.statistics analysis of variance

3.statistics arithmetic mean

4.statistics arithmetic median

5.statistics arithmetic mode

6.statistics arithmetic range

7.statistics bar graph

8.statistics best point estimation

9.statistics beta distribution

10.statistics binomial distribution

11.statistics blackscholes model

12.statistics boxplots

13.statistics central limit theorem

14.statistics chebyshevs theorem

15.statistics chisquared distribution

16.statistics chi squared table

17.statistics circular permutation

18.statistics cluster sampling

19.statistics cohens kappa coefficient

20.statistics combination

21.statistics combination with replacement

22.statistics comparing plots

23.statistics continuous uniform distribution

24.statistics cumulative frequency

25.statistics coefficient of variation

26.statistics correlation coefficient

27.statistics cumulative plots

28.statistics cumulative poisson distribution

29.statistics data collection

30.statistics data collection questionaire designing

31.statistics data collection observation

32.statistics data collection case study method

33.statistics data patterns

34.statistics deciles statistics

35.statistics dot plot

36.statistics exponential distribution

37.statistics f distribution

38.statistics f test table

39.statistics factorial

40.statistics frequency distribution

41.statistics gamma distribution

42.statistics geometric mean

43.statistics geometric probability distribution

44.statistics goodness of fit

45.statistics grand mean

46.statistics gumbel distribution

47.statistics harmonic mean

48.statistics harmonic number

49.statistics harmonic resonance frequency

50.statistics histograms

51.statistics hypergeometric distribution

52.statistics hypothesis testing

53.statistics interval estimation

54.statistics inverse gamma distribution

55.statistics kolmogorov smirnov test

56.statistics kurtosis

57.statistics laplace distribution

58.statistics linear regression

59.statistics log gamma distribution

60.statistics logistic regression

61.statistics mcnemar test

62.statistics mean deviation

63.statistics means difference

64.statistics multinomial distribution

65.statistics negative binomial distribution

66.statistics normal distribution

67.statistics odd and even permutation

68.statistics one proportion z test

69.statistics outlier function

70.statistics permutation

71.statistics permutation with replacement

72.statistics pie chart

73.statistics poisson distribution

74.statistics pooled variance r

75.statistics power calculator

76.statistics probability

77.statistics probability additive theorem

78.statistics probability multiplicative theorem

79.statistics probability bayes theorem

80.statistics probability density function

81.statistics process capability cp amp process performance pp

82.statistics process sigma

83.statistics quadratic regression equation

84.statistics qualitative data vs quantitative data

85.statistics quartile deviation

86.statistics range rule of thumb

87.statistics rayleigh distribution

88.statistics regression intercept confidence interval

89.statistics relative standard deviation

90.statistics reliability coefficient

91.statistics required sample size

92.statistics residual analysis

93.statistics residual sum of squares

94.statistics root mean square

95.statistics sample planning

96.statistics sampling methods

97.statistics scatterplots

98.statistics shannon wiener diversity index

99.statistics signal to noise ratio

100.statistics simple random sampling

101.statistics skewness

102.statistics standard deviation

103.statistics standard error se

104.statistics standard normal table

105.statistics statistical significance

106.statistics formulas

107.statistics notations

108.statistics stem and leaf plot

109.statistics stratified sampling

110.statistics student t test

111.statistics sum of square

112.statistics tdistribution table

113.statistics ti 83 exponential regression

114.statistics transformations

115.statistics trimmed mean

116.statistics type i amp ii errors

117.statistics variance

118.statistics venn diagram

119.statistics weak law of large numbers

120.statistics z table

121.discuss statistics


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