To solve this problem, we can use the Central Limit Theorem which states that the distribution of sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is large enough.
We are given that the population mean is 84 tons and the population standard deviation is 1.6 tons. Since we are sampling 34 cars at random, which is considered a large enough sample size, we can use the normal distribution to approximate the distribution of sample means.
The mean of the sampling distribution of sample means, also known as the mean of x-bar, is equal to the population mean, which is 84 tons.
The standard deviation of the sampling distribution of sample means, also known as the standard error of x-bar, is equal to the population standard deviation divided by the square root of the sample size, which is 1.6 tons / sqrt(34).
We want to find the probability that the mean load weight x is less than 83.5 tons. This can be written as P(x < 83.5).
To calculate this probability, we need to standardize the value of 83.5 using the z-score formula:
z = (x - mean) / standard deviation
z = (83.5 - 84) / (1.6 / sqrt(34))
Calculating this value, we find that z ≈ -1.2274.
Using a standard normal distribution table or a calculator, we can find the probability that z < -1.2274 is approximately 0.1101.
Therefore, the probability that 34 cars chosen at random will have a mean load weight x of less than 83.5 tons of coal is approximately 0.1101.
Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 84 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean = 84 tons and standard deviation = 1.6 ton. What is the probability that 34 cars chosen at random will have a mean load weight x of less than 83.5 tons of coal? (Round your answer to four decimal places.)
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