Question

A researcher wants to test if the mean annual salary of all lawyers in a city is different from $110,000. A random sample of 50 lawyers selected from the city reveals a mean annual salary of $ 112000, Assume that σ-$16 100, and that the test is to be made at the 2% significance level What are the critical values of z?
2.33 and 2.33
2.05 and 2.05
-1.645 and 1.645
-1.28 and 1.28
What is the value of the test statistic, z, rounded to three decimal places?
What is the p-value for this hypothesis test, rounded to four decimal places?
Should you reject or fail to reject the null hypothesis in this test?

Answer 1

Solution :

This is the two tailed test.

The null hypothesis and the alternate hypothesis is as :

Null hypothesis is $H_0:\mu=110000$

Alternate hypothesis is $H_0:\mu \neq110000$

$\overline x = 112000, \ \mu = 110000, \sigma = 16100, n = 50, \alpha = 0.02$

Therefore, the critical value of z is :

$z_{\alpha} = -2.33 \text{ and}\ 2.33$

Now the test statics is :

$z=\frac{\frac{(\overline x - \mu)}{\sigma}}{\sqrt n}$

$z=\frac{\frac{(112000-110000)}{16100}}{\sqrt {50}}$

$z=0.87$

The test statics is 0.878

We see that it is a right tailed test.

$P(z > 0.878)=1-P(z<0.878) = 1 - 0.81 = 0.19$

$P- \text{value}= \ 2 \times 0.19$

              = 0.3800

Thus , P-value > α

So we fail to reject the null hypothesis.