Final answer:
To find the proportion of adult male German Shepherds weighing more than 45 kg, calculate the z-score and use a standard normal distribution table, which is approximately 1.92%. The probability of a randomly selected adult male German Shepherd weighing between 33 kg and 39 kg is approximately 57.93%. Twenty percent of adult male German Shepherds weigh less than 33.13 kg. The sampling distribution of the average weight of 35 randomly selected adult male German Shepherds is also normally distributed with a mean of 36.4 kg and a standard deviation of 0.7117 kg. The probability of the average weight of 35 adult male German Shepherds being less than 37.5 kg is approximately 93.92%. The answers in parts (d) and (e) would still be valid under the Central Limit Theorem even if the weights of adult male German Shepherds did not follow a normal distribution, as long as the sample size is sufficiently large.
Step-by-step explanation:
(a) To find the proportion of adult male German Shepherds that will weigh more than 45 kg, we need to calculate the z-score for 45 kg using the formula: z = (x - mean) / standard deviation. The z-score for 45 kg is (45 - 36.4) / 4.2 = 2.05. Using a standard normal distribution table or a calculator, we can find that the proportion of values greater than a z-score of 2.05 is approximately 0.0192 or 1.92%.
(b) To find the probability that a randomly selected adult male German Shepherd will weigh between 33 kg and 39 kg, we need to calculate the z-scores for 33 kg and 39 kg using the formula: z = (x - mean) / standard deviation. The z-score for 33 kg is (33 - 36.4) / 4.2 = -0.81, and the z-score for 39 kg is (39 - 36.4) / 4.2 = 0.62. Using a standard normal distribution table or a calculator, we can find that the probability is approximately 0.5793 or 57.93%.
(c) To find the weight at which twenty percent of all adult male German Shepherds weigh less, we need to calculate the z-score associated with a cumulative probability of 0.2. Using a standard normal distribution table or a calculator, we find that the z-score is approximately -0.84. We can then use the z-score formula to find the corresponding weight: x = mean + (z * standard deviation) = 36.4 + (-0.84 * 4.2) = 33.13 kg.
(d) The sampling distribution of X-bar, the average weight of 35 randomly selected adult male German Shepherds, will also be normally distributed, with a mean equal to the mean of the population, 36.4 kg, and a standard deviation equal to the standard deviation of the population divided by the square root of the sample size, 4.2 kg / sqrt(35) = 0.7117 kg.
(e) 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 calculate the z-score using the formula: z = (x - mean) / (standard deviation / sqrt(sample size)). The z-score for 37.5 kg is (37.5 - 36.4) / (4.2 / sqrt(35)) = 1.56. Using a standard normal distribution table or a calculator, we can find that the probability is approximately 0.9392 or 93.92%.
(f) If the weights of adult male German Shepherds did not follow a normal distribution, the answers in parts (d) and (e) would still be valid under the Central Limit Theorem as long as the sample size is sufficiently large (usually greater than 30). The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution regardless of the shape of the population distribution.