Presentation on Data Analysis

1. Presentation on Data Analysis Z Test & T Test PRESENTED BY : GROUP 1 MD SHAHIDUR RAHMAN ROLL# 003 MD AMINUL ISLAM ROLL# 007 MRS ROZINA KHANAM ROLL# 038

2. Introduction • Sometimes measuring every single piece of item is just not practical • Statistical methods have been developed to solve these problems • Most practical way is to measure a sample of the population • Some methods test hypotheses by comparison • Two most familiar statistical hypothesis tests are : • T-test • Z-test Cont’d

3. Introduction • Z-test and T-test are basically the same • They compare between two means to suggest whether both samples come from the same population • There are variations on the theme for the T-test • Having a sample and wish to compare it with a known mean, single sample T-test is applied Cont’d

4. Introduction • Both samples not independent and have some common factor (geo location, before - after), the paired sample T-test is applied • Two variations on the two sample T-test: • The first uses samples with unequal variances • The second uses samples with equal variances

5. When to use Z-Test and T-Test • Use a Z-Test when you know the mean (µ) of the population we are comparing our sample to and the standard deviation () of the population we are comparing our sample to. • Use T-test for dependant samples when subjects tested are matched in some way or use T-test for independent samples when subjects are not matched.

6. Comparing Z-test and t-test The Z-test compares the mean from a research sample to the mean of a population. Details (μ, σ) of the population must be known. The t-test compares the means from two research samples. Used when the population details (μ, σ) are unknown.

7. T-test • A T-test is a statistical hypothesis test • The test statistic follows a Student’s T-distribution if the null hypothesis is true • The T-statistic was introduced by W.S. Gossett under the pen name “Student” • T-test also referred to as the “Student T-test” • T-test is most commonly used Statistical Data Analysis procedure for hypothesis testing • It is straightforward and easy to use • It is flexible and adaptable to a broad range of circumstances Cont’d

8. T-test • T-test is best applied when: • Limited sample size (n < 30) • Variables are approximately normally distributed • Variation of scores in the two groups is not reliably different • If the populations’ standard deviation is unknown • If the standard deviation is known, best to use Z-test Cont’d

9. T-test • Various T-tests and two most commonly applied tests are : • One-sample T-test : Used to compare a sample mean with the known population mean. • Paired-sample T-tests : Used to compare two population means in the case of two samples that are correlated. Paired sample t-test is used in ‘before after’ studies, or when the samples are the matched pairs, or the case is a control study.

10. Data types can be analysed with T-tests • Data sets should be independent from each other except in the case of the paired-sample t-test • Where n<30 the t-tests should be used • The distributions should be normal for the equal and unequal variance t-test • The variances of the samples should be the same for the equal variance t-test Cont’d

11. Data types can be analysed with T-tests • All individuals must be selected at random from the population • All individuals must have equal chance of being selected • Sample sizes should be as equal as possible but some differences are allowed

12. Sequence of T-Test (Paired Sample) Assumptions: Matched pair, normal distributions, same variance and observations must be independent of each other. Steps in the calculation: 1. Set up hypothesis: Two hypotheses H0=Assumes that mean of two paired samples = H1=Assumes that means of two paired samples 2. Select the level of significance: Normally 5% 3. Calculate the parameter: t = d /  s2 / n , n-1 is df 4. Decision making: Compare calculated value (cv) with table value (tv). If cv tv, reject H0 , If cv tv, accept H0and say that there is no significant mean difference between the two paired samples in the paired sample t-test.

13. Z-Test • The Z-test is also applied to compare sample and population means to know if there’s a significant difference between them. • Z-tests always use: • Normal distribution • Ideally applied if the standard deviation is known Cont’d

14. Z-Test • Z-tests are often applied if : • Other statistical tests like t-tests are applied in substitute • Incase of large samples (n > 30) • When t-test is used in large samples, the t-test becomes very similar to the Z-test • Fluctuations that may occur in t-tests sample variances, do not exist in Z-tests

15. Data types can be analysed with Z-test • Data points should be independent from each other • Z-test is preferable when n is greater than 30 • The distributions should be normal if n is low, if n>30 the distribution of the data does not have to be normal • The variances of the samples should be the same Cont’d

16. Data types can be analysed with Z-test • All individuals must be selected at random from the population • All individuals must have equal chance of being selected • Sample sizes should be as equal as possible but some differences are allowed

17. Z-test may ask two questions: Question #1: Does the research sample come from a population with a known mean? Example: Does prenatal exposure to drugs affect the birth weight of infants? Question #2: Is the population mean really what it is claimed to be? Examples: Does this type of car really run 12 kpl? Does this diet pill really let people lose an average of 25 pounds in 6 weeks?

18. Sequence of Z-Test (One-Sample) • Research question: Do Dhaka College students differ in IQ scores from the average college student of BD? • Data : National average,  = 114,  = 15, N=150, X = 117 • Steps in Calculation: • 1.   Set null and alternative hypothesis:(From data) • H0:  = 114 ,  mean of the population from which we got our sample is equal to 114. • H1:   114 , mean of the population from which we got our sample is not equal to 114. • 2. Select level of significance, generally 5% • 3.   State decision rules :  If zobs < +1.96 or zobs > -1.96, reject H0 • 4.   Compute standard error of mean: x = /N = 1.225 • 5.   Calculate z-value: z = X - µ / x = + 2.45 Cont’d

19. Sequence of Z-Test (One-Sample) • 6.   Compare observed z to decision rules, and make decision to reject or not reject null. • 2.45 > 1.96, so reject H0.  so, more likely that the sample mean is from some other population. • Statistically significant difference between sample mean and the population mean. • 7.   If H0 rejected, compare sample mean, and make a conclusion about the research question: • Observed mean was statistically significantly greater than the population mean we compared it to 117 > 114. • So, it can be concluded that Dhaka College students have higher IQ test scores than the average college students of BD.

20. Summary • Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s T-distribution. • A T-test is appropriate when handling small samples (n<30) while a Z-test is appropriate when handling moderate to large samples (n > 30). • T-test is more adaptable than Z-test since Z-test will often require certain conditions to be reliable. Additionally, T-test has many methods that will suit any need. • T-tests are more commonly used than Z-tests. • Z-tests are preferred than T-tests when standard deviations are known.

21. Q & A