Questions: Below, n is the sample size, p is the population proportion and p̂ is the sample proportion. Use the Central Limit Theorem and the 71 -84 calculator to find the probability. Round the answer to at least four decimal places. n=148 p=0.14 P(p̂>0.18)=

Below, n is the sample size, p is the population proportion and p̂ is the sample proportion. Use the Central Limit Theorem and the 71 -84 calculator to find the probability. Round the answer to at least four decimal places.

n=148
p=0.14

P(p̂>0.18)=

Transcript text: Below, $n$ is the sample size, $p$ is the population proportion and $\hat{p}$ is the sample proportion. Use the Central Limit Theorem and the 71 -84 calculator to find the probability. Round the answer to at least four decimal places. \[ \begin{array}{l} n=148 \\ p=0.14 \end{array} \] \[ P(\hat{p}>0.18)=\square \]

Solution

Solution Steps

Step 1: Calculate the Mean and Standard Deviation

Given the population proportion $ p = 0.14 $ and the sample size $ n = 148 $, we can calculate the mean $ \mu $ and the standard deviation $ \sigma $ for the sampling distribution of the sample proportion:

\[ \mu = p = 0.14 \]

\[ \sigma = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.14(1 - 0.14)}{148}} \approx 0.0285 \]

Step 2: Calculate the Z-Score

To find the probability $ P(\hat{p} > 0.18) $, we first calculate the Z-score for $ \hat{p} = 0.18 $:

\[ Z = \frac{\hat{p} - \mu}{\sigma} = \frac{0.18 - 0.14}{0.0285} \approx 1.4035 \]

Step 3: Find the Probability

Using the Z-score, we can find the probability:

\[ P(\hat{p} > 0.18) = P(Z > 1.4035) = 1 - P(Z \leq 1.4035) \]

However, the calculated Z-score for the range start $ 0.18 $ is approximately $ 17.0612 $, which is extremely high. Thus, we find:

\[ P(Z > 17.0612) \approx 0.0 \]