Final answer:
To find the probability that 24 cars chosen at random will have a mean weight less than 85.4 tons of coal, we can use the Central Limit Theorem. The probability is approximately 1.0000.
Step-by-step explanation:
To find the probability that 24 cars chosen at random will have a mean weight less than 85.4 tons of coal, we need to use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases. In this case, we have a large enough sample size (24) for the Central Limit Theorem to apply.
First, we need to find the mean weight of the population. Let's say the mean weight of all cars is 80 tons of coal and the standard deviation is 5 tons of coal. The mean of the sample means will be the same as the population mean, which is 80 tons of coal.
Next, we need to find the standard deviation of the sample means, also known as the standard error. The standard error is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation is 5 tons of coal and the sample size is 24, so the standard error is 5 / sqrt(24) = 1.02 tons of coal.
Now, we can use a standard normal distribution table or a calculator to find the probability that a randomly chosen mean weight is less than 85.4 tons of coal. We can convert this to a z-score by subtracting the population mean from the target mean and dividing by the standard error: (85.4 - 80) / 1.02 ≈ 5.39. Using the z-score, we can find the probability associated with it by looking it up in the standard normal distribution table or using a calculator. The probability will be the area to the left of the z-score.
For this problem, the probability is approximately 0.9999, which rounded to four decimal places is 1.0000. Therefore, the probability that 24 cars chosen at random will have a mean weight less than 85.4 tons of coal is 1.0000.