Final answer:
To find the probability that the mean of a random sample of 4 frogs lies within one gram of the true mean, we can use the concept of the sampling distribution of the mean.
Step-by-step explanation:
To find the probability that the mean of a random sample of 4 frogs lies within one gram of the true mean, we can use the concept of the sampling distribution of the mean. In this case, since the population is normally distributed with unknown mean (μ) and standard deviation (σ), we can assume that the sample means will also be normally distributed. The mean of the sampling distribution of the mean is equal to the population mean (μ), and the standard deviation of the sampling distribution of the mean (also known as the standard error) is equal to the population standard deviation (σ) divided by the square root of the sample size. In this case, since the population standard deviation (σ) is 1 gram and the sample size is 4, we can calculate the standard deviation of the sampling distribution of the mean as σ/√n = 1/√4 = 0.5 gram.
We want to find the probability that the mean of a random sample of 4 frogs lies within one gram of the true mean. This is equivalent to finding the probability that the mean lies within 0.5 gram on either side of the true mean. To find this probability, we can use the Z-score formula, which is given by Z = (X - μ) / (σ/√n), where X is the value we are interested in (in this case, X = μ ± 1), μ is the population mean, σ is the population standard deviation, and n is the sample size. Since the distribution is normal, we can use the standard normal distribution table to find the probability. We can find the Z-score for the lower bound as Z = (1 - μ) / (0.5/√4) and the Z-score for the upper bound as Z = (-1 - μ) / (0.5/√4). We can then look up the corresponding probability values in the standard normal distribution table and subtract the lower probability from the upper probability to find the probability that the mean lies within one gram of the true mean.