Answer to Question #253644 in Statistics and Probability for Goldie

Question

Based on the concept on testing the difference between two means using z-test and the learning exercises that you have done, write your arguments or lessons learned below.

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Answer 1

Z-test is a statistical test that compares the difference between two population means. Two random samples of sizes $n_1$ and $n_2$ respectively are drawn from two normal populations with means $\mu_1$ and $\mu_2$ and variances $\sigma^2_1$ and $\sigma^2_2$.

Conditions for this test to be performed include,

• Two samples must be randomly selected.
• The samples must be independent.
• Each sample size must be at least 30 and the population variances are known.

If these requirements are met, the difference of the two sample means follow a standard normal distribution with mean $\mu_{\bar{x}_1-\bar{x}_2}=\mu_1-\mu_2$ and variance, $\sigma^2_{\bar{x}_1-\bar{x}_2}=\sqrt{(\sigma_1^2/n_1)+(\sigma^2_2/n_2)}$.

The Null hypothesis to be tested is given as, $H_0:\mu_1=\mu_2$ against the alternative hypothesis that can either be two-sided or one-sided and they can be written as below.

Two sided alternative hypothesis

$H_1:\mu_1\not=\mu_2$

One sided alternative hypothesis

$H_1:\mu_1\gt\mu_2$ or $\mu_1\lt\mu_2$

The test statistic used is,

$Z_0=(\bar{x}_1-\bar{x}_2)/\sqrt{(\sigma_1^2/n_1)+(\sigma^2_2/n_2)}$ '

The null hypothesis is rejected if the calculated $Z_0$ is such that $|Z_0|\gt Z_{\alpha/2}$ for a two sided test and for a one sided test it is rejected if $Z_0\gt Z_\alpha$ for upper tailed one sided test and if $Z_0\lt-Z_\alpha$ for lower tailed one sided test for a specified $\alpha$ level of significance.