suppose that 35% of all business executives a...
suppose that 35% of all business executives are willing to switch companies if offered a higher salary. if a headhunter randomly contacts a simple random sample of 100 executives, what is the probability that over 40% will be willing to switch companies if offered a higher salary? (choose the best/closest answer to account for minor rounding)\n.1469\n.1977\n.4207\n.8023\n.8531
Answer
# Explanation:
## Step1: Identify the parameters
Let $p = 0.35$ (population - proportion), $n=100$ (sample - size). The sampling - distribution of the sample proportion $\hat{p}$ is approximately normal with mean $\mu_{\hat{p}}=p = 0.35$ and standard deviation $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.35\times(1 - 0.35)}{100}}=\sqrt{\frac{0.35\times0.65}{100}}=\sqrt{\frac{0.2275}{100}}=\sqrt{0.002275}\approx0.0477$.
## Step2: Standardize the value
We want to find $P(\hat{p}>0.40)$. First, we calculate the z - score: $z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.40 - 0.35}{0.0477}=\frac{0.05}{0.0477}\approx1.05$.
## Step3: Find the probability
Using the standard normal distribution table, $P(Z > 1.05)=1 - P(Z\leqslant1.05)$. From the standard - normal table, $P(Z\leqslant1.05) = 0.8531$. So, $P(Z > 1.05)=1 - 0.8531=0.1469$.
# Answer:
$0.1469$