1. REVIEW t-Distribution t-Distribution

2. t-Distribution The student t distribution was first derived by William S. Gosset in 1908. t is used to represent random variable. t-distribution is very commonly used in statistical inference. Like normal distribution, t-distribution is symmetrical about 0. For larger degree of freedom, the t- distribution approaches standard normal distribution.

3. t-distribution for various degree of freedoms

4. EXCEL FUNCTION for t-Distribution Given t, to find cumulative probability TDIST(t, df, tails) t:Random variable, t can not be negative df:degree of freedoms Tails:1 for one tail, 2 for two tails TDIST returns the probability for random variable >t Example: TDIST(1.5,50,1) = 0.07

5. EXCEL FUNCTION for t-Distribution Given cumulative probability, to find random variable t for two-tail test TINV(p, df) p:probability df:degree of freedoms TINV is the reverse of TDIST. TINV returns the t-value of the t-distribution as a function of the probability and the degrees of freedom. Example: TINV(0.05, 30) = 2.0423

6. Right Tail Probability One Tail Probabilities from a to  P(t>a) a

7. RIGHT TAIL PROBABILITIES One Tail P(t>a) = area between a and ∞ –Probability to the RIGHT of aEXCEL: =TDIST(a,df,1) P(t 100 >0.56) = TDIST(0.56,100,1) = 0.2884

8. RIGHT TAIL PROBABILITIES One Tail negative value a P(t<a) P(t>a)

9. RIGHT TAIL PROBABILITIES One Tail negative value –Excel does not work for negative vales of t. –But the t-distribution is symmetric. Thus, TDIST(-a,df,1) gives the area to the left of a negative value of a. 1-TDIST(-a,f,1) gives the area to the right of a negative value of a. P(t 100 >-0.56) = 1-TDIST(0.56,100,1) = 0.7116

10. Left Tail Probability One Tail Probabilities from -  to a P(t>a) a P(t<a)

11. LEFT TAIL PROBABILITIES One Tail P(t>a) = area between -∞ and a –Probability to the LEFT of aEXCEL: =1-TDIST(t,df,1) P(t 100 <0.56) = 1-TDIST(0.56,100,1) = 1-0.2884 = 0.7116

12. PROBABILITIES Two Tails EXCEL: =TDIST(a,df,2)

13. PROBABILITIES Two Tails TDIST(a,df,2)TDIST(a,df,2) gives twice the area to the right of a positive value of t. P(t>|a|) = P(t a) =P(t a) = area between –a and -∞ and area between a and ∞ EXCEL: =TDIST(a,df,2) P(t 100 >0.56) = TDIST(0.56,100,2) = 0.5767

14. Given Two-Tail Probability to find t value TINV(p,df) returns the value a, such that P(|t| > a) = probability or P(t a) = probabilityEXCEL: =TINV(P,df) t 0.05,100 = TINV(0.05,100) = 1.984

15. Given One-Tail Probability to find t value If one-tail probability is given, to find the t value, you need to multiply the probability by 2EXCEL: =TINV(2*P,df) t 0.05,100 = TINV(2*0.05,100) = 1.66

16. Probability Distribution Knowledge of the population and its parameter(s) allows us to use the probability distribution to make probability statements about the individual members of the population Population & Parameter(s) Individual Probability distribution

17. Sample Distribution Knowledge of parameter(s) and some information allows us make probability statements about sample statistic Population & Parameter(s) Statistic Sample distribution

18. Statistical Inference Sample from population and compute the required statistic to draw inferences about the parameter. Parameter(s)Statistic Sample distribution

19. REVIEW If tails = 1, TDIST (a,df,1) is calculated to determine P(t>a). Excel does not work on negative a. If tails = 2, TDIST (a,df,2) is calculated to determine P(|t| > a) = P(t > a or t < -a). If test is two-tail, TINV( ,df) returns that value a, such that P(|t| > a) = P(t a) = . If test is one-tail, TINV(2* ,df) returns that value a, such that P(t > a).