Final answer:
The probabilities for a Siamese cat's weight are found using Z-scores and the normal distribution. The mean is 4.5 kg, and the standard deviation is 0.8 kg, and calculations are adjusted when considering a sample vs. an individual cat.
Step-by-step explanation:
The probability calculations for the weights of adult male Siamese cats follow a normal distribution, where the mean (μ) is 4.5 kg and the standard deviation (σ) is 0.8 kg.
- To calculate the probability that a randomly selected cat weighs less than 4 kg, we use the Z-score formula: Z = (X - μ) / σ. Plugging the values in, Z = (4 - 4.5) / 0.8, we get Z = -0.625. Then, we look up this Z-score in the standard normal distribution table (or use a calculator) to find the corresponding probability.
- To find the probability that a cat weighs between 3.5 kg and 5 kg, we calculate the Z-scores for both weights and find the area between them in the standard normal distribution.
- For 45 randomly selected cats, we use the Central Limit Theorem which states that the distribution of the sample mean will also follow a normal distribution with mean = μ and standard deviation = σ / sqrt(n), where n is the sample size. The new Z-score is calculated with this adjusted standard deviation and the probability is found as before.
- To determine the weight of the lightest cat in the top 33%, we need to find the Z-score that corresponds to the 67th percentile of the normal distribution and then convert this Z-score back to the weight using the formula X = Zσ + μ