biostatist

Student’s t-distribution

Student’s t-distribution In cases where the population variance σ2 is unknown we can use the sample variance S2 as the best point estimate for the population variance σ2 The distribution will not follow the standard normal distribution (Z distribution), but it will follow the t-distribution

Student’s t-distribution The most important characteristics of t-distribution are :It has a mean of zeroIt is symmetric around the meanIt ranges between - ∞ - +∞4. Compared to the standard normal distribution the curve is less peaked with higher tails5. The quantity n-1 which is called the degrees of freedom (df) is used in computing the sample variance6. The t-distribution approaches the standard normal distribution as the degrees of freedom approaches infinity

Student’s t-distribution The formula for calculating the value of t: _ X-µ t =---------- s /√n

CI when population variance is unknown

Confidence Interval for the mean of a normal distribution with unknown population variance , and a small sample sizeThe reliability coefficient will be the t-value (rather than the Z-value) corresponding to the confidence level, and the degree of freedom CI=Estimator ± R.C x SE R.C= t-value t 1-α/2 , df=n-1

Confidence Interval

Confidence interval for difference between two population means when the population variances are unknown and unequal Calculation of df: (S12/n1 + S22/n2)2 df=--------------------------------- (S12/n1)2/n1 + (S22/n2)2/n2 Formula _ _CI{(X1-X2) ± t 1-α/2, df=n1+n2-2 √S12/n1+ S22/n2}

Confidence Interval

CI for the difference between two population means when the population variances are unknown but assumed to be equalWe should first find the Pooled Variance S2p (n1-1)S12+ (n2-1)S22S2p =-------------------------- n1+n2-2 _ _CI{(X1-X2) ± t 1-α/2, df=n1+n2-2 Sp√1/n1 +1/n2 }

Formula

(n1-1)S12+ (n2-1)S22S2p =-------------------------- n1+n2-2 _ _CI{(X1-X2) ± t Sp√1/n1 +1/n2} 1-α/2 df=n1+n2-2


Pairing
CI for the mean differenceMany studies are designed to produce observations in pairs .i.e.: BP, RBS….before and after giving certain treatment (Before, After) A measurement done by two instruments, individuals, times,….Every individual here has a pair of readings

Pairing

Find the difference d _Find the mean of difference dFind the standard deviation of the difference SdThe df= n-1(since we have only one sample)Apply the formula: _ CI { d± t (1-α/2, df=n-1) Sd /√n}