Answer:
The probability that the sampling distribution of sample porportions is less than 77% is P(p<0.77)=0.1106.
Step-by-step explanation:
We know that the population proportion is π=0.81.
We want to know the probability that the sampling distribution of sample proportions, with sample size n=144, is less than 0.77.
The sampling distributions of sampling proportions has a mean and standard deviation calculated as:
[tex]\mu_p=\pi=0.81\\\\\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.81\cdot 0.19}{144}}=\sqrt{0.001068}=0.0327[/tex]
Then, we calculated the z-score for p=0.77:
[tex]z=\dfrac{p-\pi}{\sigma_p}=\dfrac{0.77-0.81}{0.0327}=\dfrac{-0.04}{0.0327}=-1.2232[/tex]
The probability that the sample proportion is less than 0.77 is:
[tex]P(p<0.77)=P(z<-1.2232)=0.1106[/tex]
