According to records, the amount of precipitation in a certain city on a November day has a mean of 0.10 inches, with a standard deviation of 0.05 inches. What is the probability that the mean daily precipitation will be 0.105 inches or less for a random sample of 40 November days (taken over many years)? Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

Step 1 ：We are given the population mean (\(\mu = 0.10\) inches) and the population standard deviation (\(\sigma = 0.05\) inches). We are asked to find the probability that the sample mean (\(\bar{x}\)) is less than or equal to 0.105 inches for a sample size (\(n\)) of 40.

Step 2 ：The Central Limit Theorem tells us that for large sample sizes, the sampling distribution of the sample means approximates a normal distribution regardless of the shape of the population distribution. This allows us to use the standard normal distribution (Z-distribution) to find the probability.

Step 3 ：First, we need to calculate the standard error (SE), which is the standard deviation of the sampling distribution. The formula for standard error is \(SE = \frac{\sigma}{\sqrt{n}}\).

Step 4 ：Then, we need to calculate the Z-score, which tells us how many standard deviations an element is from the mean. The formula for Z-score is \(Z = \frac{\bar{x} - \mu}{SE}\).

Step 5 ：Finally, we can use the Z-score to find the probability (P) that the sample mean is less than or equal to 0.105 inches. We can use a Z-table to find this probability.

Step 6 ：Using the given values, we find that \(SE = 0.0079\), \(Z = 0.6325\), and \(P = 0.736\).

Step 7 ：Final Answer: The probability that the mean daily precipitation will be 0.105 inches or less for a random sample of 40 November days is approximately \(\boxed{0.736}\).