A new stockbroker (XYZ) claims that his brokerage fees are lower than that of your current stock broker's (ABC). Data available from an independent research firm indicates that the mean and std-dev of all ABC broker clients are $18 and $6, respectively. A sample of 100 clients of ABC is taken and brokerage charges are calculated with the new rates of XYZ broker. If the mean of the sample is $18.75 and std-dev is the same ($6), can any inference be made about the difference in the average brokerage bill between ABC and XYZ broker?

$n=100\\\bar x=18.75\\\sigma=6$

Hypotheses,

$H_0:\mu=18\\vs\\H_1:\mu\not=18$

The test statistic is given as,

$Z={\bar x-\mu\over {\sigma\over\sqrt{n}}}={18.75-18\over{6\over10}}={0.75\over0.6}=1.25$

The table value at $\alpha=0.05$ is $Z_{0.025}=1.96$ (It is a two sided test)

The null hypothesis is rejected if, $Z\gt Z_{0.025}$

Since $Z=1.25\lt Z_{0.025}=1.96$ we fail to reject the null hypothesis and conclude that there is insufficient evidence to infer that there is any difference between the rates of the existing broker and the new broke at 5% level of significance.