Questions: 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 0.06 inches. What is the probability that the mean daily precipitation will be 0.098 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.

Solution Steps

We are given that the mean daily precipitation in a certain city on a November day is \( \mu = 0.10 \) inches, with a standard deviation of \( \sigma = 0.06 \) inches. We want to find the probability that the mean daily precipitation for a random sample of \( n = 40 \) November days is \( 0.098 \) inches or less.

The standard error (SE) of the sample mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.06}{\sqrt{40}} \approx 0.0095 \]

To find the Z-score for the upper bound of \( 0.098 \) inches, we use the formula: \[ Z = \frac{X - \mu}{SE} \] Substituting the values: \[ Z_{end} = \frac{0.098 - 0.10}{0.0095} \approx -0.211 \]

The probability that the sample mean is less than or equal to \( 0.098 \) inches is given by: \[ P = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(-0.211) - 0 \] From the output, we find: \[ P \approx 0.417 \]

Final Answer