Registered nurses earned an average annual salary of $\backslash (69,110$. For that same year, a survey was conducted of $41$ California registered nurses to determine if the annual salary is higher than $\backslash )69,110$ for California nurses. The sample average was$\backslash (71,121$with a sample standard deviation of $\backslash )7,489.$ Conduct a hypothesis test.

$M=\$71,121,S=\$7489,n=41,a=0.05$

State the hypothesis:

The null hypothesis states that the average annual salary for registered California nurses is less than or equal to $\$69,110$. In symbols:

$H_0:\mu \le \$69,110$

The alternative hypothesis states that the average annual salary for registered California nurses is higher than $\$69,110$. In symbols:

$H_a:\mu >\$69,110$

The degree of freedom is:

$$\begin{gathered} df=n-1 \\ =41-1 \\ =40 \end{gathered}$$

From the t distribution table, for a right-tailed test with $a=0.05\text{for}df=40$, the critical value of t values or the critical region are $t=1.684$.

The test statistic is:

$$\begin{gathered} t=\frac{M-\mu }{S/\sqrt{n}} \\ =\frac{71,121-69,110}{7489\sqrt{41}} \\ =\frac{2011}{1169.585} \\ =1.72 \end{gathered}$$

We may infer that the average yearly income for registered California nurses is more than$\$69,110$ since the null hypothesis was rejected.