Hypothesis Testing with ease
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Hypothesis testing is a statistical method used to determine if there is sufficient evidence from sample data to draw conclusions about a population, involving a null hypothesis (H0) and an alternative hypothesis (H1). The significance level (alpha) helps decide whether to reject or accept the null hypothesis based on p-values, with a common threshold set at 0.05. The document also explains the Central Limit Theorem and provides examples of hypothesis testing with different distributions and sample sizes.
Hypothesis Testing with ease
- 2. Definition A hypothesis test is a statistical test that is used for determining whether there is enough evidence from the sample data to draw a conclusion for the entire population. Two types of conclusions: 1. Null Hypothesis (Ho): is the hypothesis that any observe variation in a sample is simply because of random chance variation or we can say “the hypothesis - that there is no significant difference between the sample and the population, and any observed difference is due to randomness or experimental error.” Rupak Roy
- 3. 2. Alternative Hypothesis ( Ha ): is the hypothesis testing that is contrary to the null hypothesis. Examples: If i replace the battery in my car, then my car will give better mileage? Null Hypothesis (Ho): no difference of mileage even if we replace the battery of the car. Alternative Hypothesis (Ha): difference in mileage if we replace the battery of the car Rupak Roy
- 4. Significance level i.e. alpha a If the criteria used for rejecting the null hypothesis is less than 5% i.e. 0.05(p-value) then we will conclude that there is difference between sample and population. In other words we are rejecting the null hypothesis. The most standard value for rejecting null hypothesis is 0.05; however we can change depending on our need. Rupak Roy
- 5. Example If P (value) > Significance level (a), then we will accept the null hypothesis Else P (value) < Significance level (a), then we will reject the null hypothesis Another term for saying we have rejected the null hypothesis is Statistically Significant result. Rupak Roy
- 6. Stages of Hypothesis 1. Select Null hypothesis (Ho): no difference of mileage if we replace the battery of the car. Alternative Hypothesis(Ha): difference in mileage if we replace the battery of the car. 2. Test Distribution: select appropriate distribution like norm.dist, binom.dist, t-distribution with significance level: alpha (a) 5% i.e. 0.05 3. P-value ( example, p = 1- norm.dist(………)=0.09 4. Result: failed to reject the null i.e. accepting the null hypothesis and discarding the alternative hypothesis. We will conclude that there is no difference in mileage even if we replace the battery of the car. Rupak Roy
- 7. Example A food production unit produces a particular product of an average weight of 10 lbs. with a standard deviation of 0.35 lbs. A random sample of 30 units found a slightly increase of average weight by 2 lbs. i.e. 12 lbs. So are there any issues in the product process? Significance level (a) = 0.05 Null Hypothesis (H0): There are no issues in the production process, what we found in the sample are due to random chance variation / randomness. Alternative Hypothesis (H1): There are some issues in the production process that is leading to the increase in weight per unit. Test Distribution: normal distribution Rupak Roy
- 8. Example: continued In Excel, normal distribution = norm.dist( X, mean, Standard deviation, Cumulative) where, X =12, mean = 10, standard deviation = 0.35 and cumulative = TRUE/False Therefore, = 1- norm.dist (Because we need to calculate P-value for greater than 10 lbs.) =1- norm.dist (12,10,0.35,TRUE) = 5.5089E-09 i.e. less than 0.05 Since P-value is smaller than Significance level (a), we have failed to reject the H1 i.e. accepting the alternative hypothesis and discarding the Null hypothesis. In other words, we will conclude that there are some issues in the production process that leads to the increase in weight per unit of production. Rupak Roy
- 9. Terminology Confidence level: is (1-significance level), it refers how confident you are about your conclusion. So, if null hypothesis is rejected at a 5% level of significance, then it means you are 95% (1- 0.05) confident about your conclusion. Again, if null hypothesis is rejected at a 1% level of significance, then it means you are 99% (1-0.01) confident about your conclusion. Rupak Roy
- 10. Central Limit Theorem (CLT) The central limit theorem says irrespective of the underlying population distribution, when you pick a multiple random samples from an underlying population with a sample size of at least 30 or above. The distribution of sample average will be normal even if the underlying population is not normal. Rupak Roy
- 11. Hypothesis testing when sample size is low Remember: Central limit theorem says if the sample size is sufficiently large, the distribution of sample averages will be normal irrespective of underlying population distribution or else it will follow t-distribution. So to compute the probability if the sample size is less than 30, we will use t-dist to calculate the P-value. And is also a continuous probability distribution. As we can see in the diagram when the sample size increases to 30, the t-distribution approximates a normal distribution. Rupak Roy
- 12. T-distance In order to calculate t- distribution we need t-distance i.e. the test statistics = Where, (sample mean – population mean) / ( S ) standard deviation/ (N ) sample size ) Rupak Roy
- 13. Steps for T-distribution Select null hypothesis (ho): alternative hypothesis (h1): Significance level: 5% Test distribution: t-distribution(calculate P-value) Conclusion: reject the null hypothesis or accept the null hypothesis. Rupak Roy
- 14. Example The seller of a manufacturing company claims that an average fluorescent light stays for 320 days. The inspector randomly selects 10 fluorescent lights for inspection. The sampled last with an average of 280 days along with a standard deviation of 95. What is the likelihood that the randomly selected sample fluorescent light would have an average life of no more than 280 days? Here, sample mean = 280 population mean = 320 population std. deviation = 95 sample size = 10 Rupak Roy
- 15. In excel: 1) calculate t- distance t =(280-320)/(95 / 10 ) Alternatively, (280-320)/(95/ (10^0.5)) t = - 1.331 2) use the T-distance value in Excel with the following formula = t.dist (t-distance, degrees of freedom, TRUE) = t.dist( -1.331,9,TRUE) = 0.10788 = 11% Therefore there is 11% likelihood that the average life for randomly selected bulbs is less than 280 days ALTERNATIVELY, = 1-(t.dist( t-distance , degree of freedom, TRUE)) = 1-(t.dist(-1.331,9,TRUE) = 1- 0.1078= 0.89= 89% Therefore there is 89% likelihood that the average life for 10 randomly selected bulbs is more than 280 days Note: Df = degrees of freedom = N -1 ( here in the example N (samples size) = 10) Rupak Roy
- 16. Note: Why sometimes we use 1- normal.distribution 1- t.distribution If we have notice in any distribution, cumulative for normal.distribution = norm.dist(….cumulative) where cumulative is TRUE / FALSE TRUE (function) means < and FALSE (function) = point probability And what if we want > there is no function, so for that we manually have to feed 1 – appropitate.distribution Rupak Roy
- 17. What if population Std.deviation is not available If population standard deviation is not known, sample deviation can be substitute for the population standard deviation. Therefore, S =sample deviation / sample size Rupak Roy
- 18. What if population distribution is not normal i.e. not normal distribution? We are using normal distribution to calculate p-value for hypothesis testing but it is not always necessary that every hypothesis test must use a normal distribution. If we already know the type of distribution, then it’s better to use directly the right distribution for hypothesis testing. Remember the example from our previous slide “Stage of Hypothesis” where in point number 2 we have mentioned that we can choose any appropriate types of distribution. Rupak Roy
- 19. Recap: “Stages of Hypothesis” 1. Select Null Hypothesis (Ho): no difference of mileage if we replace the battery in the car. Alternative Hypothesis (Ha): difference in mileage if we replace the battery in the car 2. Test Distribution: select appropriate distribution like norm.dist, binom.dist with significance level: alpha (a) 5% 3. P-value ( example, p = 1- norm.dist(………) )=0.09 4. Result: failed to reject the null i.e. accepting the null hypothesis and discarding the alternative hypothesis. We will conclude that there is no difference in mileage even if we replace the battery of the car. Rupak Roy
- 20. Next Directional Hypothesis test like one tail test i.e. if you have strong reason to believe in your hypothesis. And more. Rupak Roy
- 21. To be continued. Rupak Roy