Hypothesis testing basics in the field of statistics

This presentation has the basics about hypothesis testing in statistics

1. HYPOTHESIS TESTING IT IS A METHOD FOR TESTING A CLAIM OR HYPOTHESIS ABOUT A PARAMETER IN A POPULATION, USING DATA MEASURED IN A SAMPLE. IN THIS METHOD, WE TEST SOME HYPOTHESIS BY DETERMINING THE LIKELIHOOD THAT A SAMPLE STATISTIC COULD HAVE BEEN SELECTED, IF THE HYPOTHESIS REGARDING THE POPULATION PARAMETER WERE TRUE.

2. Normality test No rmality tests  are used to determine if a  data set  is well-modeled by a  normal distribution  and to compute how likely it is for a  random variable  underlying the data set to be normally distributed. If the variable is normally distributed, we use parametric test that are based on this assumption. If a variable fails a normality test, in that case we use non-parametric test.

3. PARAMETRIC TEST Parametric tests assume a normal distribution of values, or a “bell-shaped curve.” Parametric tests are in general more powerful (require a smaller sample size) than  nonparametric tests . Example :- height   if you were to graph height from a group of people, one would see a typical bell-shaped curve. Height is roughly a normal distribution.

4. PARAMETRIC TEST CONTI . The most widely used tests are the t-test (paired or unpaired), ANOVA (one-way, two-way) 

5. WHAT IS A PAIRED T-TEST Paired t-test (also known as a dependent or correlated t-test) is a statistical test that compares the averages/means and standard deviations of two related groups to determine if there is a significant difference between the two groups. A significant difference occurs when the differences between groups are unlikely to be due to sampling error or chance. The groups can be related by being the same group of people, the same item, or being subjected to the same conditions. Paired t-tests are considered more powerful than unpaired t-tests because using the same participants or item eliminates variation between the samples that could be caused by anything other than what’s being tested.

6. WHAT ARE THE HYPOTHESES OF A PAIRED T-TEST? There are two possible hypotheses in a paired t-test. The  null hypothesis   states that there is no significant difference between the means of the two groups. The  alternative hypothesis   states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

7. WHAT ARE THE ASSUMPTIONS OF A PAIRED T-TEST? The dependent variable is normally distributed The observations are sampled independently The dependent variable is measured on an incremental level, such as ratios or intervals. The independent variables must consist of two related groups or matched pairs.

8. WHEN TO USE A PAIRED T-TEST? Paired t-tests are used when the same item or group is tested twice, which is known as a repeated measures t-test. Examples:- The before and after effect of a pharmaceutical treatment on the same group of people. Body temperature using two different thermometers on the same group of participants. Standardized test results of a group of students before and after a study prep course.

9. WHAT IS AN UNPAIRED T-TEST? An unpaired t-test (also known as an independent t-test) is a statistical procedure that compares the averages/means of two independent or unrelated groups to determine if there is a significant difference between the two.

10. WHAT ARE THE HYPOTHESES OF AN UNPAIRED T-TEST? The hypotheses of an unpaired t-test are the same as those for a paired t-test. The two hypotheses are: The null hypothesis states that there is no significant difference between the means of the two groups. The alternative hypothesis states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

11. WHEN TO USE AN UNPAIRED T-TEST? An unpaired t-test is used to compare the mean between two independent groups. We use an unpaired t-test when we are comparing two separate groups with equal variance. Examples Research during which there are two independent groups, such as women and men, that examines whether the average bone density is significantly different between the two groups. Comparing the average commuting distance traveled by residents of two states using 1,000 randomly selected participants from each city.

12. PAIRED VS UNPAIRED T-TEST The key differences between a paired and unpaired t-test are summarized below. A paired t-test is designed to compare the means of the same group or item under two separate scenarios. An unpaired t-test compares the means of two independent or unrelated groups. In an unpaired t-test, the variance between groups is assumed to be equal. In a paired t-test, the variance is not assumed to be equal.

13. ANOVA ( ANALYSIS OF VARIANCE ) It is a statistical technique that is used to check if the means of two or more groups are significantly different from each other. ANOVA checks the impact of one or more factors by comparing the means of different samples. Example :- We can use ANOVA to prove/disprove if all the medication treatments were equally effective or not.

14. ONE-WAY ANOVA A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

15. EXAMPLES OF WHEN TO USE A ONE WAY ANOVA Situation 1:  You have a group of individuals randomly split into smaller groups and completing different tasks. For example, you might be studying the effects of tea on weight loss and form three groups: green tea, black tea, and no tea. Situation 2:  Similar to situation 1, but in this case the individuals are split into groups based on an attribute they possess. For example, you might be studying leg strength of people according to weight. You could split participants into weight categories (obese, overweight and normal) and measure their leg strength on a weight machine.

16. LIMITATIONS OF THE ONE WAY ANOVA A one way ANOVA will tell you that at least two groups were different from each other. But  it won’t tell you which groups were different.

17. TWO-WAY ANOVA A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable.  With a two-way ANOVA, there are two independents. Example:- A two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.

18. ASSUMPTIONS FOR TWO WAY ANOVA The  population  must be close to a  normal distribution . Samples  must be independent. Population  variances  must be equal. Groups must have equal  sample sizes .

19. NON-PARAMETRIC TESTS It is used when continuous data are not normally distributed or when dealing with discrete variables.  N on-parametric tests  are designed for real data: skewed, lumpy, having a outliers, and gaps scattered around.

20. WILCOXON SIGNED RANK TEST The Wilcoxon signed rank test should be used if the differences between pairs of data are non-normally distributed. used to determine two dependent samples selected from population having the same distribution. similar to paired T test in parametric tests

21. CHI-SQUARE TEST: Chi-Square Test is used to examine the association between two or more variables measured on categorical scales. Chi-Square is used most frequently to test the statistical significance of result reported in bivariate tables, and interpreting bivariate tables is integral to interpreting the results of a chi-square test.

22. Fisher’s exact TEST: It is the substitute for chi square test with small or imbalance data sets.

23. P- Value The p - value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. A p value is the probability of obtaining a sample outcome, given that the value stated in the null hypothesis is true. The p value for obtaining a sample outcome is compared to the level of significance. The p - value is used as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected.

24. P VALUE CONTI. A p value is used in hypothesis testing to help you support or reject the null hypothesis. The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. On the other hand, a large p-value means your results have a huge probability of being completely random and not due to anything in the experiment. Therefore, the smaller the p-value, the more important (“significant”) the results are.

25. ONE-TAILED TEST A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis . Alpha levels  (sometimes just called “significance levels”) are used in  hypothesis tests ;

26.  A one-tailed test has the entire 5% of the alpha level in one tail (in either the left, or the right tail). 

27. TWO TAILED TEST A two-tailed test allots half of your alpha to testing the statistical significance in one direction and half of your alpha to testing statistical significance in the other direction. Two tailed test allots half of your alpha to testing the statistical significance in one direction and half of your alpha to testing statistical significance in the other direction.  This means that .025 is in each tail of the distribution of your test statistic.

28. Two-tailed hypothesis tests are also known as nondirectional and two-sided tests because you can test for effects in both directions. When you perform a two-tailed test, you split the significance level percentage between both tails of the distribution. 

29. Z-TEST A z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. T he test statistic is assumed to have a normal distribution, and nuisance parameters such as standard deviation should be known in order for an accurate z-test to be performed. A z-statistic, or z-score, is a number representing how many standard deviations above or below the mean population a score derived from a z-test is. Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size. Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known.

30. Z test Conti. For each  significance level  in the  confidence interval , the  Z -test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the  Student's  t -test  whose critical values are defined by the sample size (through the corresponding  degrees of freedom ).

31. ONE-SAMPLE Z TEST We perform the One-Sample Z test when we want to compare a sample mean with the population mean. Example :- Let’s say we need to determine if girls on average score higher than 600 in the exam. We have the information that the standard deviation for girls’ scores is 100. So, we collect the data of 20 girls by using random samples and record their marks. Finally, we also set our ⍺ value (significance level) to be 0.05.

32. Since the P-value is less than 0.05, we can reject the null hypothesis  and conclude based on our result that Girls on average scored higher than 600.

33. TWO SAMPLE Z TEST We perform a Two Sample Z test when we want to compare  the mean of two samples. Example Here, let’s say we want to know if Girls on average score 10 marks more than the boys. We have the information that the standard deviation for girls’ Score is 100 and for boys’ score is 90. Then we collect the data of 20 girls and 20 boys by using random samples and record their marks. Finally, we also set our ⍺ value (significance level) to be 0.05.

34. Thus, we can  conclude based on the P-value that we fail to reject the Null Hypothesi s. We don’t have enough evidence to conclude that girls on average score of 10 marks more than the boys.

35. DECIDING BETWEEN Z TEST AND T-TEST If the sample size is large enough, then the Z test and t-Test will conclude with the same results.  For a  large sample size ,  Sample Variance will be a better estimate  of Population variance so even if population variance is unknown, we can  use the Z test using sample variance. Similarly, for a  Large Sample , we have a high degree of freedom. And since t -distribution approaches the normal distribution ,  t he difference between the z score and t score is negligible.

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