History of corporate acquisitions. Refer to the Academy of Management Journal (August 2008) investigation of the performance and timing of corporate acquisitions, Exercise 2.12 (p. 74). Recall that the investigation discovered that in a random sample of 2,778 firms, 748 announced one or more acquisitions during the year 2000. Does the sample provide sufficient evidence to indicate that the true percentage of all firms that announced one or more acquisitions during the year 2000 is less than 30%? Use $\mathit{a}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$to make your decision.

The number of firms sampled during the year 2000, n= 2778.

The number with acquisitions during the year 2000, x= 748.

The significance level,$a=0.05$.

The condition required for a valid large sample hypothesis test is given as follows:

⦁ A random sample is selected from a binomial population.

⦁ The sample size n is large (The condition will be satisfied if both $n{p}_0=\ge 15$ and $n{q}_0=\ge 15$)

The sample proportion is calculated as:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{748}{2778} \\ =0.27 \end{gathered}$$

The null hypothesized proportion is calculated as:

$$\begin{gathered} p_0=\frac{30}{100} \\ =0.3 \end{gathered}$$

The null hypothesis is that the true percentage of all firms that announced one or more acquisitions during the year 2000 is not less than 30%.

i.e.$H_0:p=0.3$

The alternative hypothesis is that the percentage of all firms that announced one or more acquisitions during the year 2000 is less than 30%.

i.e; $H_a:p<0.3$

The significance level is,$a=0.05$.

The test statistic is computed as follows:

$$\begin{gathered} Z_c=\frac{(\hat{p}-p_0)}{\sqrt{{\displaystyle \frac{p_0{q}_0}{n}}}} \\ =\frac{(0.27-0.3)}{\sqrt{\left({\displaystyle \frac{0.3(1-0.3)}{2778}}\right)}} \\ =-3.45 \end{gathered}$$

This is a one-tailed hypothesis. The $Z_a$value for $a=0.05$is calculated from the standard normal table is 1.645.

The rejection region is $Z_c<-Z_a\Rightarrow -3.45<-1.645$. So; we reject the null hypothesis.

Hence, there is enough evidence to claim that the percentage of all firms that announced one or more acquisitions during the year 2000 is less than 30%.