Third-Party. In 2010, shortly before that year's midterm elections, when Americans were dissatisfied with government and elections when Americans were dissatisfied with the government and the Tea Party movement was emerging as a political force, $58\%$of American adults thought that a third major party was needed. In a recent Gallup poll, $60\%$of $1028$Americans felt the same way, At the $10\%$significance level, do the data provide sufficient evidence to conclude that the percentage of all American adults who now think that a third major party is needed has changed from that in$2010$ ?

The significance level is $1\%$

Number of Americans$n=1028$

A number of American adults thought that a third major party is needed $p_0=0.58$

Calculate the value of$n{p}_0$

$n{p}_0=\left(1028\right)(0.8)$

$=596.24$

Calculate the value of

$n\left(1-p_0\right)=\left(1028\right)(1-0.58)$

$=431.76$

Both these values are greater than $5$.

So one proportion $z-$test is appropriate to use.

The null hypothesis is

$H_0:p_0=0.58$

The alternate hypothesis is

$H_0:p_0\ne 0.58$

The expression for $z$is

$z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$

The $z$value is calculated as

$z=\frac{0.60-0.58}{\sqrt{\frac{0.58(1-0.58)}{1028}}}$

$=\frac{0.02}{0.0154}$

$=1.30$

Also $\alpha =0.10$

The test is two-tailed

For $\alpha =0.05$, the critical value of $z$from the standard table is

$z_{0.05}=\pm 1.645$

The test statistic is within acceptable bounds. As a result, the hypothesis $H_0$is not ruled out.

At the $10\%$level, the test results are not statistically significant.

As a result, the data does not provide adequate evidence to infer that the percentage of all persons in the United States who now believe a third major party is required has changed since $2010$.