Question

Question: Independent random samples n₁ =233 and n₂=312 are selected from two populations and used to test the hypothesis ${\mathit{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{(}{\mathbf{\mu }}_{\mathbf{1}}\mathbf{-}\mathbf{\mu }\mathbf{)}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$against the alternative ${\mathit{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{(}{\mathbf{\mu }}_{\mathbf{1}}\mathbf{-}\mathbf{\mu }\mathbf{)}}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$

.a. The two-tailed p-value of the test is 0.1150 . Interpret this result.b. If the alternative hypothesis had been ${\mathit{H}}_{\mathbf{a}}\mathbf{:}{\left({\mu }_1-\mu \right)}_{\mathbf{2}}\mathbf{<}\mathbf{0}$ , how would the p-value change? Interpret the p-value for this one-tailed test.

Short answer

Answer

Random sampling guarantees that the findings received from your sample are close to those received if the full sample was surveyed.

Step-by-step solution

(a) Interpret the result for the given two-tailed p-value of the test

The null hypothesis is $H_a:{\left({\mu }_1-\mu \right)}_2=0$

If we take the significance level to be 0.05 and the p-value is 0.1150.

As the p-value is more than the significance level, so the null hypothesis will not be rejected.

If we take the significance level to be 0.01 , and the p-value is 0.1150 .

As the p-value is more than the significance level, so the null hypothesis will not be rejected.

(b) Interpret the p-valueHa:(μ1-μ)2<0  .

If the alternative hypothesis has changed from $H_a:{\left({\mu }_1-\mu \right)}_2\ne 0$to$H_a:{\left({\mu }_1-\mu \right)}_2<0$ , it implies that the p-value should be halved as now it is a one-tailed test instead of a two-tailed test.

So, the p-value will be 0.0575 .