Answer:
0.01596
Step-by-step explanation:
A scientist claims that 8% of the viruses are airborne
Given that:
The population proportion p = 8%
The sample size = 477
We can calculate the standard deviation of the population proportion by using the formula:
[tex]\sigma_p = \sqrt{\dfrac{p(1-p)}{n}}[/tex]
[tex]\sigma_p = \sqrt{\dfrac{0.8(1-0.8)}{477}}[/tex]
[tex]\sigma_p = \sqrt{\dfrac{0.0736}{477}}[/tex]
[tex]\sigma_p = 0.02098[/tex]
The required probability can be calculated as:
[tex]P(| \hat p - p| > 0.03) = P(\hat p - p< -0.03 \ or \ \hat p - p > 0.03)[/tex]
[tex]= P \bigg ( \dfrac{\hat p -p }{\sqrt{\dfrac{p(1-p)}{n}}} < -\dfrac{0.03}{0.0124} \bigg ) + P \bigg ( \dfrac{\hat p -p }{\sqrt{\dfrac{p(1-p)}{n}}} >\dfrac{0.03}{0.0124} \bigg )[/tex]
= P(Z < -2.41) + P(Z > 2.41)
= P(Z < -2.41) + P(Z < -2.41)
= 2P( Z< - 2.41)
From the Z-tables;
[tex]P(| \hat p - p| > 0.03)[/tex] = 2 ( 0.00798
[tex]P(| \hat p - p| > 0.03)[/tex] = 0.01596
Thus, the required probability = 0.01596
