Statistical tests of significance and Student`s T-Test

Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained

1. SUBMITTED BY - VASUNDHRA KAKKAR M. PHARMACY (PHARMACEUTICS) SEMESTER- 3 STATISTICAL TESTS OF SIGNIFICANCE AND STUDENT`s T- TEST

2. Statistical tests of Significance A test of significance is a procedure to compare observed data with a claim also called hypothesis , the truth of which is being assessed. The claim is a statement about a parameter, like the population proportion (p) or the population mean (µ). The results of a significance test are expressed in terms of a probability that measures how well the data and the claim agree. Statistical tests are performed to determine statistical significance of results.

3. Steps in Testing of Statistical Significance 1) State the Null Hypothesis and Alternative hypothesis 2) Select a alpha level of significance 3)Find the P-value (using a table or any statistical software) 4) Compare P-value with α- value and decide whether the null hypothesis should be rejected or accepted 5) Interpret the result by stating if the results are statistically significant or not.

4. Null and Alternative Hypothesis The null hypothesis (denoted by H ) states the value of a population parameter (such as proportion, or mean) is equal to some claimed value. The term null is used because this hypothesis assumes that there is no difference between the two means or the recorded difference is not significant. The alternative hypothesis (denoted by Ha) states the value of population parameter is not equal to some claimed value. The alternative hypothesis is the claim that researchers are actually trying to prove is true.

5. Null and Alternative Hypothesis Example - Null hypotheses states : Hypothesized Population mean (µ ) for weight of tablets is100 mg which is equal to population mean (µ). So, H : µ = 100 Alternative hypothesis states : Population mean (µ) is significantly different from µ (100mg). So, Ha : µ > 100 or µ < 100 or µ ≠ 100 If , Ha : µ > 100 then Right- Tailed test is performed. If , Ha : µ < 100 then Left- Tailed test is performed. If, Ha : µ ≠ 100 then Two- Tailed test is performed.

6. ALPHA LEVEL OF SGNIFICANCE The significance level ( α ) the probability of making type 1 error (rejecting the null hypothesis when it was in fact true). The term significance level  is referred to as pre-chosen probability. As this quantity represents an error rate, lower values are generally preferred. Its values generally ranges from 0.05 to 0.10.  

7. P-VALUE P- value is the calculated chance that type 1 error has occurred. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. When the p- value is less than 0.01, the result is called highly significant. If we choose significance level of 0.05,  p -value less than 0.05 (typically ≤ 0.05) will be statistically significant. A   p -value higher than 0.05 (> 0.05) is not considered statistically significant.

8. TYPES OF SIGNIFICANCE TEST PARAMETRIC TESTS NON-PARAMETRIC TESTS Student`s t- test Analysis of variance (ANOVA) Regression Correlation Z - test Chi- squared test Wilcoxan Signed- Rank test Kruskal -Wallis test

9. Used for drawing conclusions or interpretations for small samples. (I) Application of t-test to assess the significance difference between the sample mean and population mean : The calculation of t-value involves the following steps: ( i ) Null Hypothesis : Null hypothesis (Ho): sample mean (X̅) = population mean (µ) or Ho = X̅ = µ (ii) Test statistics : T-value is calculated by the following formula: STUDENT`s T- TEST

10. X̅ = Sample mean S = standard deviation of sample n = number of unit in sample (iii) Degree of freedom: It is one less than the number of units in the sample. Degree of freedom (d.f. or v) = No. of units in the sample – 1 ( n – 1) (iv) Level of significance: Any level of significance can be considered to test the hypothesis but generally 1 % (0.01 ) or 5% ( 0.05) levels of probability is considered for testing the hypothesis. (v) Table value of t: After calculating the t- value with the above formula, the tabulated value of t is noted from Fishers and Yates table at given degree of freedom and 5% level of significance. Then calculated value of t is compared with the tabulated value of t.

11. (vi) Test criterion . If the calculated value of t irrespective of (+) or (-) sign is less than the tabulated value, then the Null hypothesis is accepted. But, if the calculated value of t is greater than the tabulated value of t the null hypothesis is rejected. (II) T-test to assess the significance difference between the means of two samples drawn from the same population: The procedure of this test is as follows: ( i ) Null hypothesis: H = µ 1   = µ 2 where µ 1  and µ 2 are the standard deviations of the sample I and sample II respectively.

12. (ii) Test statistics: Next, the value of t is calculated by the following formula: (iii) Degree of freedom: (d.f.) = n 1 + n 2  – 2 (iv) Level of significance: The level of significance is generally considered at 5% (0.05) level of probability. (v) Tabulated value of t: Value of t is recorded from the Fisher and Yates table at the given degree of freedom and at 5% level of significance. (vi) Test criterion: At last, the calculated value of t is compared with the table value of t. If the calculated value of t exceeds the table value, the observed difference is considered statistically significant and hence null hypothesis is rejected.

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