Question

In the upcoming governor's election, the most recent poll based on 900 respondents predicts that the incumbent will be reelected with 55% of the votes. for the sake of argument, assume that 51% of the actual voters in the state support the incumbent governor (p = 0.51). calculate the probability of observing a sample proportion of voters 0.55 or higher supporting the incumbent governor.

Answer 1

Solution: We are given:

$\hat{p}=0.55,p=0.51,n=900$

We have to find $P(\hat{p}\geq 0.55)$

Now we have to find the z score corresponding to $\hat{p}=0.55$

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

        $=\frac{0.55-0.51}{\sqrt{\frac{0.51(1-0.51)}{900} } }$

        $=\frac{0.04}{0.0167}$

        $=2.40$

$\therefore P(\hat{p}\geq 0.55}=P(z\geq 2.4)$

Using the standard normal table, we have:

$P(z\geq 2.4)=0.0082$

Therefore, the probability of observing a sample proportion of voters 0.55 or higher supporting the incumbent governor is 0.0082

Answer 2

Answer:  

0.0081

Explanation:

Let p be the population proportion of voters in the state support the incumbent governor .

According to the given information , we have

Population proportion : $p=0.51$

Sample size (Number of respondents): n= 900

Sample proportion of voters in the state support the incumbent governor : $\hat{p}=0.55$

Further  Explanation:

Formula to find z-score (for proportion) :  

$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

$z=\dfrac{0.55-0.51}{\sqrt{\dfrac{0.51(1-0.51)}{900}}}=2.40048014405\approx2.40$ [Rounded to the nearest two decimal places]

To find the probability of observing a sample proportion of voters 0.55 or higher supporting the incumbent governor , we find the p-value below :

$P(p>0.55)=P(z>2.40)\ \ \text{[Area on under the right side of the normal curve greater than z=2.40]}\\\\=1-P(z<2.40)\ \ [\because P(Z>z)=1-P(Z<z)]\\\\=1- 0.9918024\ \ [\text{By using online z-value calculator}]\\\\=0.0081976\approx0.0081$ [Rounded to the nearest 4 decimal places.]

Therefore , the probability of observing a sample proportion of voters 0.55 or higher supporting the incumbent governor = 0.0081

Refer the given attachment for further understanding.

Learn more :

• https://brainly.com/question/13716713  [Answered by Jeanashupp]

Keywords :

sample proportion , z-value , z-table , Normal curve.