1. Sumukh Deshpande n Lecturer College of Applied Medical Sciences Statistics = Skills for life. BIOSTATISTICS (BST 211) Lecture 9

2. The t-test Introduction & Applications

3. Confidence Intervals … so far  Data is N(  )   is KNOWN  Calculate the Confidence Interval:  What if…   is UNKNOWN?  Data is NOT N(  )?  How do we Calculate the Confidence Interval?

4. Data is N(  ) but  UNKNOWN  If n ≥ 30, you may use s instead of  and continue as before.

5. Data is N(  ) and  UNKNOWN  If n < 30, but the population is NORMALLY distributed… Then, the t-Distribution can be used to estimate  with a given confidence level

6. What is the t-Distribution?  Symmetrical about the mean, like N.  It has a mean = 0, like N  It has a larger variance depending on df’s.  Every df has its own t curve  The t-distribution is broader and fatter than N.  As df increases, the t- distribution approaches N.  Total area under the curve = 1

7. t-distribution with various df’s

8.  Comparison of t dist with various df to Normal dist

9. Reading the t-Table This is a TWO tails table

10. Confidence using t  Similar to N(  )  Need df  Confidence level  One or 2 tails?  Take t from the table

11. Quick Example 20 Bagaa watermelons have an average mass of 6.5 kg and a SD of 1.6 kg. Estimate the mean mass for the population with 95% confidence level?  What do we know?  n = 20, xbar = 6.5, s = 1.6 and  = 5%  What shall we use z or t?  One tail or two tails?

12. Quick Example  t (19, 0.05)2 = 2.0930   = 2.093(1.6/ √20)   = 0.749   = 6.5  0.749   = 6.5  0.8  5.7 kg ≤  ≤ 7.3 kg  n = 20, xbar = 6.5, s = 1.6 and  = 5%  Two tails t-table  df = 20 -1 = 19

13. Use of t-test for Hypothesis Testing 1  A syrup is sold in bottles marked120 ml. A sample of 16 bottles have a mean of 118 ml with 3.5 ml SD.  Can we say that the content is less than 120 ml with 95% CL?  n = 16,  = 120, xbar = 118, s = 3.5,  = 5%.  2 tails or 1 tail?  What is H 0 ? H 1 ?

14. Use of t-test for Hypothesis Testing 2  n = 16,  = 120, xbar = 118, s = 3.5,  = 5%.  1 tail because we want to know if  < 120 ml  H 0 :  = 120 ml  H 1 :  < 120 ml  this is Calculated t  Compare with t-Critical.  Reject H 0 if Calc-t > t-Crit

15. Use of t-test for Hypothesis Testing 3  Calc-t = (120 – 118)/(3.5/4) = 2.2857  t (15, 0.05)1 = 1.7531  2.2857 > 1.7531  Reject H 0  We can say that  < 120 ml with 95% certainty.

16. 5 Steps for Hypothesis Testing with t  State H 0 and H 1  Establish level of significance   One tail or two tails?  Calculate Calc-t and read t-Crit  If Calc-t > t-Crit, Reject H 0 ; otherwise FAIL TO REJECT H 0.