Question

according to a survey, 10% of americans are afraid to fly. suppose 1,100 americans are sampled. round answers to 4 decimal places.
a. what is the probability that 121 or more americans in the survey are afraid to fly?
b. what is the probability that 143 or more americans in the survey are afraid to fly?
c. what is the probability that 8% or less of the americans surveyed answered they were afraid to fly?

Explanation:

Step1: Identify distribution parameters

This is a binomial - to - normal approximation problem. Let $n = 1100$ (sample size) and $p=0.1$. The mean of the binomial distribution is $\mu = np=1100\times0.1 = 110$, and the standard deviation is $\sigma=\sqrt{np(1 - p)}=\sqrt{1100\times0.1\times(1 - 0.1)}=\sqrt{1100\times0.1\times0.9}=\sqrt{99}\approx9.95$.

Step2: Standardize for part a

For $x = 121$, the z - score is $z=\frac{x-\mu}{\sigma}=\frac{121 - 110}{9.95}=\frac{11}{9.95}\approx1.11$. We want $P(X\geq121)$, which is equivalent to $P(Z\geq1.11)$ in the normal approximation. Using the standard normal table, $P(Z\geq1.11)=1 - P(Z < 1.11)=1 - 0.8665 = 0.1335$.

Step3: Standardize for part b

For $x = 143$, the z - score is $z=\frac{x-\mu}{\sigma}=\frac{143 - 110}{9.95}=\frac{33}{9.95}\approx3.32$. We want $P(X\geq143)$, which is equivalent to $P(Z\geq3.32)$ in the normal approximation. Using the standard normal table, $P(Z\geq3.32)=1 - P(Z < 3.32)=1 - 0.9996 = 0.0004$.

Step4: Standardize for part c

The proportion $\hat{p}=0.08$. The mean of the sampling distribution of the sample proportion is $\mu_{\hat{p}}=p = 0.1$ and the standard deviation is $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.1\times(1 - 0.1)}{1100}}=\sqrt{\frac{0.09}{1100}}\approx0.0090$. The z - score for $\hat{p}=0.08$ is $z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.08 - 0.1}{0.0090}=\frac{- 0.02}{0.0090}\approx - 2.22$. We want $P(\hat{p}\leq0.08)$, which is equivalent to $P(Z\leq - 2.22)$ in the normal approximation. Using the standard normal table, $P(Z\leq - 2.22)=0.0132$.

Answer:

a. $0.1335$
b. $0.0004$
c. $0.0132$