Standard error of difference between two means

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STANDARD ERROR OF DIFFERENCE BETWEEN TWO MEANS : 

STANDARD ERROR OF DIFFERENCE BETWEEN TWO MEANS - Dr. Prateek Bobhate

STANDARD ERROR : 

STANDARD ERROR It is a measure of chance variation & does not mean an error or mistake SE of mean is the SD of the means in a sampling distribution. It tells us how much variability can be expected among means in future samples.

APPLICATIONS : 

APPLICATIONS It is used very often in medical practice such as:- By means of SE, means of a normally distributed variable in two like or unlike groups are compared in terms of their height, weight, pulse etc. Eg:- Action of a drug on a variable such as BP or pulse rate is compared in two groups when a placebo is given to a control group.

Calculation of standard error of difference: : 

Calculation of standard error of difference: S.E. of difference is denoted as s(x1 – x2) or σ(x1 – x2). Following formulae are used for its calculation: 1. s(x1 – x2)= SD1 2 + SD22 n1 n2 It is applied when μ and σ of the population are unknown and sample size is large. Z = x1 – x2 s(x1 – x2)

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2. S( x1 – x2) = σ 1 + 1 n1 n2 It is applied when σ is known but μ is unknown. Z = (x1 – x2) σ 1 + 1 n1 n2

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3. SE = SD2 + SD2 n1 n2 = SD 1 + 1 n1 n2 It is applied when population variance is unknown. Hence combined variance of samples is calculated as – S2 = ∑(x1 – x1)2 + ∑(x2 – x2)2 n1 + n2 - 2

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Z = x1 – x2 s 1 + 1 n1 n2 Lets see a capsule for the same………………….

Example: : 

Example: In a study on growth of children, one group of 100 children had a mean height of 60 cm and SD of 2.5cm while another group of 150 children had a mean height of 62 cm and SD of 3 cm. Is the difference between the two groups statistically significant? SOLUTION:- Since σ and μ are not known and sample >30, applying the 1st formula – s(x1– x2)= SD1 2 + SD2 2 n1 n2

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= (2.5)2 + 32 100 150 = 0.1225 = o.35 Z = X1 – X2 = 60-62 s(X1- X2) 0.35 = -5.71 …………………. A

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Similarly we can use 3rd formula for calculation of SE of difference between two means using combined variance of two samples. S12 = ∑ ( X1 - X1 )2 n1 – 1 S22 = ∑ ( X2 – X2 )2 n2 - 1

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After substituting the values we get ; Combined Variance = sum of square of both samples n1 + n2 -2 = 618.75 + 1341 99 + 149 = 7.8961 Now, SD = 7.8961 = 2.81

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SE of difference = SD x 1 + 1 n1 n2 = 2.81 x 1 + 1 100 150 = 0.3628 Z = Observed difference = 60 - 62 SE 0.3628 = - 5.47 ……………………… B

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Thus the observed difference calculated by both the methods is more than three times the Standard Error hence it is highly significant. Thus the growth is more in 2nd group than in 1st.

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THANKS