Final answer:
The sampling distribution of X represents the distribution of sample means from multiple random samples. The probability of the average weight can be found using z-scores and the standard normal distribution. The answers in parts (a) and (b) would still be valid if the weights did not follow a normal distribution with a large enough sample size.
Step-by-step explanation:
(a) Sampling distribution of X:
The sampling distribution of X represents the distribution of the sample means taken from multiple random samples of the same size (in this case, 35 male adult German Shepherds) from a population. The mean of the sampling distribution of X is equal to the population mean (36.4 kg), and the standard deviation of the sampling distribution of X is equal to the population standard deviation divided by the square root of the sample size (4.2 kg / sqrt(35)).
(b) Probability of the average weight:
To find the probability that the average weight of 35 randomly selected adult male German Shepherds will be less than 37.5 kg, we need to convert this value into a z-score. The formula for calculating a z-score is given by z = (X - μ) / (σ / sqrt(n)), where X is the value to be converted, μ is the population mean, σ is the population standard deviation, and n is the sample size. Once we have the z-score, we can use a standard normal distribution table or calculator to find the corresponding probability.
(c) Validity of answers:
If the weights of adult male German Shepherds did not follow a normal distribution, the answers in parts (a) and (b) would still be valid as long as the sample size is large enough (which is typically satisfied with a sample size greater than 30) and the sampling distribution of X can be approximated as a normal distribution by the Central Limit Theorem.