Significance Test: One sample t- Test (Single Sample t- Test) Dr. Amjad El-Shanti 2012

One sample t- test By this test we will study whether the sample mean consistent with a certain hypothesized value for the population mean. The method used is known as significance testing based on the normal distribution if the sample size large and the t-distribution for small sample size.

One sample t- test & Paired t- test The paired t-test is a special case of the one sample t-test whether a sample mean is different from some specific value µ, which need no be zero.

One-sample t-test Formula The general formula: SE= S/√n

S = ∑Xi2 – (∑Xi)2 n √ n-1

Example (1): The following are the heights in centimeters of 24 two –year old boys with homozygous sickle cell disease (SS): Height and weight standard for the UK give a reference height for two-Year old males of 86.5 cm. Does the above sample suggest that two-year old males SS children differ in height from the standards? Answer H0: µ= 86.5 Ha: µ = 86.5 84.4 89.9 89 81.9 87 78.5 84.1 86.3 80.6 80 81.3 86.6 84.3 89.8 85.4 85 82.5 80.7 85.5

Answer example (1): cont…. X-x (X-x )2 84.4 0.3 0.09 89.9 5.8 33.64 89 4.9 24.01 81.9 -2.2 4.84 87 3.4 11.56 78.7 -5.6 31.36 84.1 86.3 2.2 80.6 -3.5 12.25 80 -4.1 16.81 81.3 -2.8 7.84 86.8 2.7 7.29 83.4 -0.7 0.49 89.8 5.7 32.49 85.4 1.3 1.96 85 0.9 0.81 82.5 -1.6 2.56 80.7 -3.4 84.3 0.2 0.04 1.69 85.5 1.4 ∑x= 2018.4 ∑(x-x)2=225.9 X= ∑X/n= 2018.4/24= 84.1cm S=√ ∑(x-x)2/n-1= = √ 225.9/23 = 3.13 Se= s/√n = 3.13/ √ 24= 0.63 X= 84.1, µ= 86.5 Se= 0.63 n=24 d . f.= 23 t test = x-µ/se= 84.1-86.5/0.63 t test = -3.81 α=0.05 , d . f. = 23 t table= -2.06 T test > t table 3.81 > 2.06 P<α ------- p< 0.05 Reject Ho There is significant difference in heights of SS children and height of Two –year children

Example (2): A sample of size n =100 has a mean age x= 54.85, Test whether this sample mean comes from a population whose mean age is 53, given that =5.5? Answer H0: µ= 53 Ha: µ = 53 α= 0.05 Z test = (x - µ)/ (/√ n)= (54.85-53)/ (5.5/√100)= 3.36 Z table (α=0.05) = 1.96 Z test > Z table ………………… 3.36 >1.96 Reject Ho …………………… There is significant difference between sample mean age and population mean age.