Short Answer 
Understanding the distribution of the housing values is crucial, specifically noting a mean of $403,000 and a standard deviation of $278,000. To calculate the probability that the mean value of a sample of 40 houses is less than $500,000, a Z-score of 2.2067 is calculated, resulting in a probability of 98.63%.
Step 1: Understand the Distribution
Begin by noting the important parameters of the distribution you are working with. In this case, you have a mean of $403,000 and a standard deviation of $278,000. Understanding these parameters is crucial because they will help you in calculating probabilities related to the sample mean of the housing values.
Step 2: Calculate the Z-Score
Next, you need to determine the Z-score, which helps you find the probability of the sample mean. For the sample of 40 houses and a threshold of $500,000, calculate the Z-score using the formula: Z = (X – ≈ú) / (œE / ‚Äöaon). Here, you find that Z = (500000 – 403000) / 43955.66 = 2.2067. This value is essential for finding the corresponding probability.
Step 3: Determine the Probability
Finally, use the Z-score to find the probability from the standard normal distribution. Look up Z = 2.2067 in a Z-table or use statistical software to get the probability. You will find that P(Z < 2.2067) = 0.9863, indicating that there is a 98.63% probability that the mean value of the 40 houses is less than $500,000.