POLI 443 Applied Political Research Session 5: Tests of Hypotheses – The Student t-test Lecturer: Prof. A. Essuman-Johnson , Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College of Education School of Continuing and Distance Education 2014/2015 – 2016/2017
The t-Test (Student t test) • Introduction Tests involving the normal distribution assume that the sample size being dealt with is 30 or more. When the sample size is less than 30 the distribution does not follow the normal distribution but the small sample distribution which is called the t distribution. • This test was the outcome of the work of a man called Gosset who published his work under the pen name Student . Slide 2
• One basic assumption of the normal distribution is that for samples of size larger than 30 i.e. N>30 or large samples, the sampling distribution of many statistics will approximately be normal. However in situations where the sample size is less than 30 i.e. N<30 the sampling distribution does not approximate to the normal distribution. This is what Gosset found out and what is known as the small sampling test. The t-test (student t) is therefore a test of Slide 3
• statistical significance used in situations where the number of cases in the sample is less than 30. The computation of t is exactly the same as that of Z as follows: Z = x̅ - µ/ ( x̅ ) = x̅ - µ/s √ N -1 Slide 4
Illustration • The mean weekly income of 18 workers at the GTP factory is GH¢72 compared to an average weekly income of GhC66 for all workers in the textile industry in Ghana. The standard deviation of the incomes for the GTP workers is GH¢4 per week. Would you say the mean income of GTP workers is higher than the mean income of all textile workers in Ghana? Test at the 0.05 level of significance? Slide 5
Solution • Elements given: µ = 66 x̅ = 72 S = 4 N = 18 • H O : That all workers in Ghana earn the same weekly income i.e. µ= 66 • H 1 : That the mean weekly income of GTP workers is higher than the rest of textile workers in Ghana. i.e. µ ≠ 66 (Two tail test) • Test Statistic = Z = x̅ - µ/ ( x̅ ) • Since is unknown we estimate it by use of s therefore we lose 1 df Slide 6
• Therefore ( x̅ ) = S/ √ N - 1 • = 4/ √ 18 - 1 • = 4/ √ 17 • = 4/4.123 • = 0.97 • Z = x̅ - µ/ ( x̅ ) • = 72 - 66/0.97 • = 6/0.97 • = 6.2 Slide 7
• Decision Rule: • At the 5% level Accept H O if Z C (Z calculated) lies within – t.975 to t.975 (two tail test) for which 18-1 = 17 df is the interval – 2.11 to 2.11. (From t tables) Reject otherwise. Reject Ho Accept Ho Reject Ho -2.11 +2.11 6.2 Slide 8
• Conclusion: Since the Z score = 6.2>2.11 we reject H O and state that GTP workers on average receive higher wages than textile workers in general. Slide 9
Summary In this section, you have learned how to do tests about small samples that do not follow the normal distribution in what is called the student t distribution. Slide 10
THANK YOU Slide 11
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