To solve this problem, we need to calculate the probability $P(0.50 < \hat{p} < 0.60)$ given the sample size $n = 111$ and the population proportion $p = 0.54$. We can use the Central Limit Theorem for proportions to approximate this probability.

Here’s a step-by-step approach:

Step 1: Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of $\hat{p}$ (also known as the standard error) is given by: $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$ Substitute $p = 0.54$ and $n = 111$: $\sigma_{\hat{p}} = \sqrt{\frac{0.54 \times (1 - 0.54)}{111}} = \sqrt{\frac{0.54 \times 0.46}{111}}$

Calculating this: $\sigma_{\hat{p}} \approx \sqrt{\frac{0.2484}{111}} \approx \sqrt{0.0022387} \approx 0.0473$

Step 2: Convert the Probability Range to Z-Scores

Now we convert $\hat{p} = 0.50$ and $\hat{p} = 0.60$ to z-scores using the formula: $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$

For $\hat{p} = 0.50$: $z = \frac{0.50 - 0.54}{0.0473} \approx \frac{-0.04}{0.0473} \approx -0.8458$

For $\hat{p} = 0.60$: $z = \frac{0.60 - 0.54}{0.0473} \approx \frac{0.06}{0.0473} \approx 1.2681$

Step 3: Use Z-Table or Calculator to Find Probabilities

Now, we look up these z-scores in the standard normal distribution table or use a calculator to find the probabilities:

1. For $z = -0.8458$, the cumulative probability $P(Z < -0.8458) \approx 0.1997$.
2. For $z = 1.2681$, the cumulative probability $P(Z < 1.2681) \approx 0.8970$.

Step 4: Calculate the Probability

The probability $P(0.50 < \hat{p} < 0.60)$ is the difference between these two cumulative probabilities: $P(0.50 < \hat{p} < 0.60) = P(Z < 1.2681) - P(Z < -0.8458)$ $P(0.50 < \hat{p} < 0.60) \approx 0.8970 - 0.1997 = 0.6973$

Answer

The probability $P(0.50 < \hat{p} < 0.60)$ is approximately 0.6973.

Would you like further details or have any questions?

Further Questions for Exploration:

1. How does the sample size affect the standard error in this context?
2. Can we approximate this probability if the sample size is much smaller, say $n = 30$?
3. What would happen to this probability if the population proportion $p$ were closer to 0.5?
4. How do we interpret this probability in the context of real-world data?
5. Can this approach be applied if $p$ is close to 0 or 1, and what are the limitations?

Tip

When dealing with sampling distributions, always verify that the sample size is large enough to justify using the Central Limit Theorem for proportions.