1. Student ’ s t-distribution

2. In cases where the population variance σ 2 is unknown we can use the sample variance S 2 as the best point estimate for the population variance σ 2 The distribution will not follow the standard normal distribution (Z distribution), but it will follow the t- distribution

3. Student ’ s t-distribution The most important characteristics of t-distribution are : 1.It has a mean of zero 2.It is symmetric around the mean 3.It ranges between - ∞ - +∞ 4. Compared to the standard normal distribution the curve is less peaked with higher tails 5. The quantity n-1 which is called the degrees of freedom (df) is used in computing the sample variance 6. The t-distribution approaches the standard normal distribution as the degrees of freedom approaches infinity

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

6. CI when population variance is unknown Confidence Interval for the mean of a normal distribution with unknown population variance, and a small sample size The 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

7. Confidence Interval Confidence interval for difference between two population means when the population variances are unknown and unequal Calculation of df: (S 1 2 /n 1 + S 2 2 /n 2 ) 2 df=--------------------------------- (S 1 2 /n 1 ) 2 /n 1 + (S 2 2 /n 2 ) 2 /n 2 Formula _ _ CI{(X 1 -X 2 ) ± t 1-α/2, df=n1+n2-2 √S 1 2 /n 1 + S 2 2 /n 2 }

8. Confidence Interval CI for the difference between two population means when the population variances are unknown but assumed to be equal We should first find the Pooled Variance S 2 p (n 1 -1)S 1 2 + (n 2 -1)S 2 2 S 2 p =-------------------------- n 1 +n 2 -2 _ _ CI{(X 1 -X 2 ) ± t 1-α/2, df=n1+n2-2 S p √1/n 1 +1/n 2 }

9. Formula (n 1 -1)S 1 2 + (n 2 -1)S 2 2 S 2 p =-------------------------- n 1 +n 2 -2 _ _ CI{(X 1 -X 2 ) ± t S p √1/n 1 +1/n 2 } 1-α/2 df=n1+n2-2

10. Pairing CI for the mean difference Many 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

11. Pairing Find the difference d _ Find the mean of difference d Find the standard deviation of the difference S d The df= n-1(since we have only one sample) Apply the formula: _ CI { d± t (1-α/2, df=n-1) S d /√n}