At the Heathrow Airport in London, England, an airport administrator believes that the proportion of delayed flights is 0.26 per day. What is the probability of selecting a sample of 50 flights on a particular day and finding the proportion is more than 0.40? Apply the Central Limit Theorem and find and interpret the probability in the exercise. 

Note: Round z-value to 2 decimal places and final answer to 4 decimal places.

To solve this problem using the Central Limit Theorem (CLT), we need to apply it to the sampling distribution of proportions. Here's the step-by-step approach:

### Step 1: Define parameters
- The population proportion \( p = 0.26 \) (the believed proportion of delayed flights).
- The sample size \( n = 50 \).
- We are asked to find the probability that the sample proportion \( \hat{p} \) exceeds 0.40.

### Step 2: Apply the Central Limit Theorem
The CLT states that for large enough sample sizes, the sampling distribution of the sample proportion \( \hat{p} \) is approximately normally distributed with:
- Mean \( \mu_{\hat{p}} = p = 0.26 \).
- Standard deviation \( \sigma_{\hat{p}} \) (standard error of the sample proportion) is given by:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.26(1 - 0.26)}{50}} = \sqrt{\frac{0.26 \times 0.74}{50}} 
\]

### Step 3: Calculate the standard deviation \( \sigma_{\hat{p}} \)
Now, I'll calculate the standard deviation.The standard deviation of the sample proportion \( \hat{p} \) is approximately \( 0.0620 \).

### Step 4: Calculate the z-score
To find the probability that the sample proportion exceeds 0.40, we need to compute the z-score:

\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.40 - 0.26}{0.0620}
\]

Now, let's calculate the z-score.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you'd like, I can walk you through the rest of the calculations manually. Would you like me to continue explaining the steps? Let me know if you have any other questions.

Here are 5 related questions for further exploration:
1. How does the Central Limit Theorem apply to different sample sizes?
2. What is the significance of the z-score in statistics?
3. How can we interpret confidence intervals in relation to population proportions?
4. Can you calculate the probability for a smaller or larger sample size?
5. What are some practical applications of the normal distribution in quality control?

**Tip:** When applying the Central Limit Theorem, remember that the sample size should be sufficiently large (typically \( n \geq 30 \)) for the approximation to hold well.

At the Heathrow Airport in London, England, an airport administrator believes that the proportion of delayed flights is 0.26 per day. What is the probability of selecting a sample of 50 flights on a particular day and finding the proportion is more than 0.40? Apply the Central Limit Theorem and find and interpret the probability in the exercise. Note: Round z-value to 2 decimal places and final answer to 4 decimal places.

Learn how to calculate the probability that more than 40% of a sample of 50 flights are delayed at Heathrow Airport, using the Central Limit Theorem. This detailed solution walks through the key concepts of sampling distributions, z-scores, and statistical probability. Ideal for advanced high school or college-level students.

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