91影视

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:

猞� A random sample is selected from a binomial population.

猞� The sample size n is large (The condition will be satisfied if both $n{p}_0=鈮�15$ and $n{q}_0=鈮�15$)

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 $Z_c<-Z_a鈬�-3.45<-1.645$. 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%.