Question

On May 16, 1994, Newsweek reported the results of a public opinion poll that asked: "From everything you know about Bill Clinton, does he have the honesty and integrity you expect in a president?" (p. 23). Poll surveyed 518 adults and 233, or 0.45 of them (clearly less than half), answered yes. Could Clinton's adversaries conclude from this that only a minority (less than half) of the population of Americans thought Clinton had the honesty and integrity to be president? Solution

Answer 1

Let p denotes the true proportion of people who have honesty and integrity as a president.

To test against

Here

sample proportion $\hat{p} = 0.45$

and sample size

The test statistic can be written as

$z = \frac{(\hat{p} - 0.5)}{\sqrt{\frac{0.5*(1-0.5)}{n}}}$ which under H₀ follows a standard normal distribution.

We reject H₀ at 5% level of significance if P-value < 0.05

Now,

The value of the test statistic $z_{obs}=-2.27596$

and P-value $=P(Z<z_{obs})=P(Z<-2.27596)=\Phi(-2.27596) = 0.0114242$

Since P-value < 0.05, so we reject H₀ at 5% level of significance and we can conclude that the proportion of people who have honesty and integrity as a president is significantly less than 50%.