Unknown and unequal variance when we can assume that

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Unknown and unequal variance When we can assume that the two populations are normally distributed and that the unknown population variances are unequal, an approximate t-test based on independent random samples is given by: t = (X ̅ 1 − X ̅ 2 )−(μ 1 −μ 2 ) ( s 1 2 n 1 + s 2 2 n 2 ) 1/2 In this formula, we use the tables of the t- distribution using the ‘modified’ degrees of freedom. The ‘modified’ degrees of freedom are calc ulated using the following formula: df = ( s 1 2 n 1 + s 2 2 n 2 ) 2 (s 1 2 /n 1 ) 2 n 1 + (s 2 2 /n 2 ) 2 n 2
Quantitative Methods 2019 Level I High Yield Notes © IFT. All rights reserved 42 Hypothesis tests concerning mean differences (Note: This section has high difficulty and low probability of being tested. Do this section once you have mastered all remaining topics) If the samples of the populations whose means we are comparing are dependent, then the paired comparison test is used. The hypothesis is structured as the difference between means of two populations. H 0 : μ d = μ d0 H a : μ d ≠ µ d0 where: μ d stands for the population mean difference and μ d0 stands for the hypothesized value for the population mean difference, which is usually zero. In order to arrive at the test statistic, we first determine the sample mean difference using: d ̅ = 1 n d i n i=0 The standard error of the mean difference as follows: s d ̅ = s d √n Once we have these two values, we can calculate the test statistic using a t-test. This is calculated using the following formula using n - 1 degrees of freedom: t = d ̅ −μ d0 s d ̅ The value of calculated test statistic is compared with the t-distribution values in the usual manner to arrive at a decision on our hypothesis. Hypothesis tests concerning variance (Note: This section has high difficulty and low probability of being tested. Do this section once you have mastered all remaining topics) Single population variance In tests concerning the variance of a single normally distributed population, we use the chi- square test statistic, denoted by χ 2 . After drawing a random sample from a normally distributed population, we calculate the test statistic using the following formula with n - 1 degrees of freedom: χ 2 = (n−1)(s 2 ) σ 0 2 where: n = sample size s = sample variance We then determine the critical values using the level of significance and degrees of
Quantitative Methods 2019 Level I High Yield Notes © IFT. All rights reserved 43 freedom. The chi-square distribution table is used to calculate the critical value. Two population variance In order to test the equality or inequality of two variances, we use an F-test which is the ratio of sample variances. The formula for the test statistic of the F-test is: F = s 1 2 s 2 2 where: 𝑠 1 2 = the sample variance of the first population with n observations 𝑠 2 2 = the sample variance of the second population with n observations A convention is to put the larger sample variance in the numerator and the smaller sample variance in the denominator.

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