Final answer:
The proportion of adult male German Shepherds weighing more than 45 kg is approximately 1.92%. The probability that a randomly selected adult male German Shepherd will weigh between 33 kg and 39 kg is approximately 41.92%. Twenty percent of all adult male German Shepherds have a weight less than approximately 32.23 kg. The sampling distribution of the average weight of 35 randomly selected adult male German Shepherds is normally distributed. The probability that the average weight of 35 randomly selected adult male German Shepherds will be less than 37.5 kg is approximately 88.49%. The validity of the answers in parts (d) and (e) depends on the assumption that the weights of adult male German Shepherds follow a normal distribution.
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. Substituting in the given values, we have z = (45 - 36.4) / 4.2 = 2.05. By looking up the z-score in the z-table, we can find the corresponding proportion, which is approximately 0.0192. Therefore, approximately 1.92% of adult male German Shepherds will weigh more than 45 kg.
(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 both values. Using the same formula as (a), we find that z1 = (33 - 36.4) / 4.2 = -0.8 and z2 = (39 - 36.4) / 4.2 = 0.62. By looking up the z-scores in the z-table, we can find the corresponding proportions. The probability is then the difference between the two proportions, which is approximately 0.4192 or 41.92%.
(c) To find the weight less than which 20% of all adult male German Shepherds fall, we need to find the z-score that corresponds to a cumulative probability of 0.2. By looking up the z-score in the z-table, we find that the corresponding z-score is approximately -0.84. Using the same formula as (a), we can solve for x: -0.84 = (x - 36.4) / 4.2. Solving for x, we find that x ≈ 32.23. Therefore, 20% of adult male German Shepherds have a weight less than 32.23 kg.
(d) The sampling distribution of X, the average weight of 35 randomly selected adult male German Shepherds, is normally distributed with a mean equal to the population mean (36.4 kg) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (√35).
(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 / √sample size). Substituting in the given values, we have z = (37.5 - 36.4) / (4.2 / √35) ≈ 1.20. By looking up the z-score in the z-table, we find that the corresponding probability is approximately 0.8849 or 88.49%.
(f) If the weights of adult male German Shepherds did not follow a normal distribution, the answers to parts (d) and (e) would not be valid. This is because the normal distribution assumption is essential for obtaining reliable results using these calculations.