Question

Random samples of size $n_1=400$ and$n_2=500$ were drawn from populations 1 and 2, respectively. The samples yielded${\text{x}}_1=105$ and $x_2=140$. Test$H_0:\left(p_1-p_2\right)⩾0$ against$H_a:\left(p_1-p_2\right)<0$at the 1% level of significance.

Short answer

At a 1% significance level, we do not have sufficient evidence to conclude that the difference between the two population proportions is less than 0.

Step-by-step solution

Given information

We have

The size of the samples is$n_1=400$and$n_2=500$

And

The number of successes is and

The sample proportion of successes is$x_1=105$and$x_2=140$

$\begin{array}{rcl}{\hat{p}}_1& =& \frac{105}{400}\\ & =& 0.2625\end{array}$

And

$\begin{array}{rcl}{\hat{p}}_2& =& \frac{140}{500}\\ & =& 0.28\end{array}$

The sample proportion

For the large sample size (at least 30), the distribution of the sample proportion is approximately normal as per the central limit theorem.

The sample proportion can be viewed as means of the number of successes per trial in the respective samples, so the Central Limit Theorem applies when the sample sizes are large.

Calculating the test statistic

The test statistic for testing these hypotheses is

$$\begin{gathered} Z=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}} \\ =\frac{0.2625-0.28}{\sqrt{\frac{0.2625\left(1-0.2625\right)}{400}+\frac{0.28\left(1-0.28\right)}{500}}} \\ =\frac{-0.0175}{\sqrt{0.000484+0.0004032}} \\ =\frac{-0.0175}{0.02979} \\ =-0.59 \end{gathered}$$

Calculating the critical value

Here

$\alpha =.01$

Using the standard normal table, the critical value at the 1% significance level and left-tailed test (alternative hypothesis is left-tailed) is -2.326

That is

$\begin{array}{rcl}{Z}_{\alpha }& =& Z_{0.01}\\ & =& -2.326\end{array}$

Decision for the null hypothesis

We can see that

$Z>Z_{\alpha }$

That is,$-0.59>-2.326$

Hence, we failed to reject the null hypothesis.

Conclusion

At a 1% significance level, we do not have sufficient evidence to conclude that the difference between the two population proportions is less than 0.