Final answer:
To find specific probabilities for weights of adult St. Bernard dogs, we calculate z-scores for the values and consult the standard normal distribution. The probability for a dog to weigh between 146 and 159 pounds is 65.5%, and for a dog to weigh less than 142 pounds is 21.2%. A dog in the 75th percentile weighs 157 pounds.
Step-by-step explanation:
The weight of adult St. Bernard dogs follows a Normal distribution with a mean weight of 150 pounds and a standard deviation of 10 pounds. To calculate the probability of a randomly selected dog weighing between 146 and 159 pounds, we convert these weights to z-scores and use the standard normal distribution.
For 146 pounds, the z-score is (146 - 150) / 10 = -0.4. For 159 pounds, the z-score is (159 - 150) / 10 = 0.9. Using a z-table or normal distribution calculator, the probability of being between these z-scores is approximately 0.6554 (65.5%).
To find the probability that the dog weighs less than 142 pounds, we calculate the z-score: (142 - 150) / 10 = -0.8. The probability corresponding to this z-score is approximately 0.2119 (21.2%).
For the 75th percentile, we find the z-score that corresponds to this percentile and then convert it back to an actual weight using the mean and standard deviation. The z-score for the 75th percentile is approximately 0.674. So the weight is 150 + (0.674 * 10) = 156.7, rounded to the nearest whole number, is 157 pounds.