When the variances are known and the sample size is high, a z-test is used to assess if two population means vary. The mean of the two players is not the same.
<h3>What is a Z-test?</h3>When the variances are known and the sample size is high, a z-test is used to assess if two population means vary.
In order to execute an appropriate z-test, the test statistic is expected to have a normal distribution, and nuisance factors such as standard deviation should be known.
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
This corresponds to a two-tailed test, and a z-test for two means, with known population standard deviations, will be used.
(2) Rejection Region
Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is
The rejection region for this two-tailed test is R={z:∣z∣>1.96}
(3) Test Statistics
The z-statistic is computed as follows:
(4.) The decision about the null hypothesis
Since it is observed that |z| = 2.475 > z_c = 1.96, it is then concluded that the null hypothesis is rejected.
Using the P-value approach: The p-value is p = 0.0133p=0.0133, and since p = 0.0133 < 0.05p=0.0133<0.05, it is concluded that the null hypothesis is rejected.
(5.) As it is concluded that the null hypothesis H₀ is rejected. Therefore, there is enough evidence to claim that the population mean μ1 is different than μ2.
Hence, the mean of the two players is not the same.
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