Answer to Question #206162 in Statistics and Probability for ELLA

Question

The operations manager of a sales company wants to see whether there is a significant difference in the ages of male and female customers. He selected a sample of 35 samples for each group. The ages are shown in the table below: Male Female x̄1 = 27.3 x̄ 2 = 28 σ1 = 2.4 σ2 = 3.1 n1 = 35 n2 = 35 At a = 0.05, decide if there is enough evidence to reject the claim of no difference in the ages of the two groups. Solution: Step 1: Null and alternative hypothesis H0: H1: Step 2: Significance level a = Step 3: Test statistic: 

Answer 1

Step 1:

The following null and alternative hypotheses need to be tested:

$H_0:\mu_1=\mu_2$

$H_1:\mu_1\not=\mu_2$

Step 2:

This corresponds to a two-tailed test, and a z-test for two means, with known population standard deviations will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}$

Step 3:

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}_1-\bar{x}_2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}$$

$$=\dfrac{27.3-28}{\sqrt{2.4^2/35+3.1^2/35}}\approx-1.05632$$

Since it is observed that $|z|=1.05632<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu_1$ is different than $\mu_2,$ at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=2P(Z>1.05632)=0.292057,$ and since $p=0.292057>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population mean $\mu_1$ is different than $\mu_2,$ at the $\alpha=0.05$ significance level.