QUESTION 8 The number of gallons of paint that Home Depot sells in a given day is normally distributed with a mean of 130 gallons and a standard deviation of 30 gallons (I realize that the distribution is probably different for weekends compared to weekdays, but just assume everyday has the distribution). What is the probability that on a randomly chosen day, Home Depot sells more than 175 gallons of paint? (please round your answer to 4 decimal places)

To find the probability that Home Depot sells more than 175 gallons of paint, we first need to standardize the value using the z-score formula: \[ z = \frac{(X - \mu)}{\sigma} \] where: - \( X = 175 \) (the value we want to find the probability for) - \( \mu = 130 \) (the mean) - \( \sigma = 30 \) (the standard deviation) Calculating the z-score: \[ z = \frac{(175 - 130)}{30} = \frac{45}{30} = 1.5 \] Now, we need to find the probability corresponding to \( z = 1.5 \). Using the standard normal distribution table or a calculator, we find: \[ P(Z < 1.5) \approx 0.9332 \] To find the probability that Home Depot sells more than 175 gallons, we compute: \[ P(Z > 1.5) = 1 - P(Z < 1.5) \] \[ P(Z > 1.5) = 1 - 0.9332 = 0.0668 \] Thus, the probability that Home Depot sells more than 175 gallons of paint on a randomly chosen day is approximately **0.0668** when rounded to four decimal places.