To solve this problem, we can use the Central Limit Theorem (CLT), which states that the distribution of the sample mean of a large enough sample size approaches a normal distribution, regardless of the shape of the population distribution.

Step 1: Identify the relevant information

• Mean (μ) of the population: 1.99 days
• Standard deviation (σ) of the population: For an exponential distribution, the standard deviation is equal to the mean, so σ = 1.99 days.
• Sample size (n): 138 shipments
• Target mean (x̄): 1.82 days

Step 2: Calculate the standard error (SE) of the sample mean

The standard error for the sample mean is given by the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma = 1.99$ (standard deviation)
• $n = 138$ (sample size)

$SE = \frac{1.99}{\sqrt{138}} \approx \frac{1.99}{11.74} \approx 0.1694$

Step 3: Standardize the target mean (1.82 days) using the z-score formula

The z-score formula is:

$z = \frac{x̄ - \mu}{SE}$

Where:

• $x̄ = 1.82$
• $\mu = 1.99$
• $SE = 0.1694$

$z = \frac{1.82 - 1.99}{0.1694} \approx \frac{-0.17}{0.1694} \approx -1.0035$

Step 4: Find the probability corresponding to the z-score

Now, we need to find the probability that the sample mean is less than 1.82 days, which is equivalent to finding the probability that the z-score is less than -1.0035.

Using the standard normal distribution table (or a calculator), we can find the cumulative probability for $z = -1.0035$.

The cumulative probability corresponding to $z = -1.0035$ is approximately 0.1587.

Final Answer:

The probability that the average shipping time is less than 1.82 days is approximately 0.1587.

Would you like more details or have any further questions?

Here are some related questions you might find useful:

1. How would the probability change if the sample size were smaller or larger?
2. What happens if the distribution is not exponential but normal?
3. How does the central limit theorem work in approximating the normal distribution?
4. How do you calculate probabilities for other types of distributions (e.g., Poisson or binomial)?
5. What are some real-world applications of using the Central Limit Theorem?

Tip: When dealing with the central limit theorem, always remember that the sample size should be sufficiently large (typically n > 30) for the approximation to be accurate.