Step 1 :The null hypothesis (H0) is that the proportion of 'a's has not changed, i.e., it is still 0.12. The alternative hypothesis (Ha) is that the proportion of 'a's has changed, i.e., it is not 0.12. So, the correct hypotheses are: \[H_{0}: p=0.12\] \[H_{a}: p \neq 0.12\]
Step 2 :Next, we calculate the z-test statistic using the formula: \[z = \frac{{\hat{p} - p_{0}}}{{\sqrt{\frac{{p_{0} * (1 - p_{0})}}{{n}}}}}\] where \(\hat{p}\) is the sample proportion, \(p_{0}\) is the assumed population proportion under the null hypothesis, and n is the sample size. In this case, \(\hat{p} = \frac{{60}}{{600}} = 0.10\), \(p_{0} = 0.12\), and n = 600. Plugging these values into the formula gives: \[z = \frac{{0.10 - 0.12}}{{\sqrt{\frac{{0.12 * (1 - 0.12)}}{{600}}}}} = -2.04\]
Step 3 :The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. For a two-tailed test (because Ha: p ≠ 0.12), we find the probability of observing a z-score less than -2.04 or greater than 2.04. This gives a p-value of 0.041.
Step 4 :Finally, we compare the p-value to the level of significance (0.05). If the p-value is less than the level of significance, we reject the null hypothesis. In this case, 0.041 < 0.05, so we reject H0. This means we believe the proportion of 'a's has changed in modern times. So, the correct conclusion is: \[\boxed{\text{Reject } H_{0}. \text{The proportion of 'a's is significantly different from 0.12.}}\]