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Z Test & T Test PRESENTED BY : GROUP 1 MD SHAHIDUR RAHMAN ROLL# 003 MD AMINUL ISLAM ROLL# 007 MRS ROZINA KHANAM ROLL# 038 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 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 Ttest Having a sample and wish to compare it with a known mean, single sample T-test is applied Cont’d 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 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. 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. A T-test is a statistical hypothesis test The test statistic follows a Student’s Tdistribution 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 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 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. Data sets should be independent from each other except in the case of the paired-sample ttest 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 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 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 H0 and say that there is no significant mean difference between the two paired samples in the paired sample t-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 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 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 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 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? 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 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. Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s Tdistribution. 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. Q&A