An airline operate from Melbourne to japan. the flight time, in hours, of each of these flights is a normally distributed random variable T with a mean of 11 hours and a standard deviation of 0.73 hours. Assume that the flight times are independent of one another. A)what proportion of flights have a flight time greater than 11.5 hours? give answer to 4 decimal places

The flight times are normally distributed with a mean \( \mu = 11 \) hours and a standard deviation \( \sigma = 0.73 \) hours.

We are asked to find the proportion of flights where the flight time \( T \) is greater than 11.5 hours. To do this, we need to calculate the probability:

\[
P(T > 11.5)
\]

### Step 1: Standardize the normal variable
First, we standardize the normal variable \( T \) to a standard normal distribution \( Z \) by using the formula:

\[
Z = \frac{T - \mu}{\sigma}
\]

Substitute \( T = 11.5 \), \( \mu = 11 \), and \( \sigma = 0.73 \):

\[
Z = \frac{11.5 - 11}{0.73} = \frac{0.5}{0.73} \approx 0.6849
\]

### Step 2: Find the corresponding probability
Now, we need to find the probability that \( Z \) is greater than 0.6849. Using standard normal distribution tables or a calculator, we find the cumulative probability \( P(Z \leq 0.6849) \).

The cumulative probability for \( Z = 0.6849 \) is approximately:

\[
P(Z \leq 0.6849) \approx 0.7535
\]

Thus, the probability of \( Z > 0.6849 \) is:

\[
P(Z > 0.6849) = 1 - P(Z \leq 0.6849) = 1 - 0.7535 = 0.2465
\]

### Final Answer:
The proportion of flights with a flight time greater than 11.5 hours is approximately:

\[
\boxed{0.2465}
\]

Would you like more details or have any questions? Here are 5 related questions to explore:

1. What is the probability that a flight lasts less than 10 hours?
2. How would the result change if the standard deviation increased to 1 hour?
3. What percentage of flights take between 10.5 and 12 hours?
4. If the airline wants 95% of flights to arrive in under a certain time, what should that time be?
5. How does changing the mean flight time affect the proportion of longer flights?

**Tip:** Always check if the problem involves a standard normal distribution; it's the foundation for many probability calculations!

An airline operates from Melbourne to Japan. The flight time, in hours, of each of these flights is a normally distributed random variable T with a mean of 11 hours and a standard deviation of 0.73 hours. Assume that the flight times are independent of one another. A) What proportion of flights have a flight time greater than 11.5 hours? Give answer to 4 decimal places.

Solve a statistics problem involving flight times on a route from Melbourne to Japan. This example uses normal distribution with a mean of 11 hours and a standard deviation of 0.73 hours to find the proportion of flights that last longer than 11.5 hours. Learn how to standardize a normal variable and apply cumulative probability to find the result. Suitable for Grades 11-12 students studying probability.

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