statistics and probility

Question

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 90 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean =90 tons and standard deviation =3 ton.

 

B) What is the probability that 12 cars chosen at random will have mean weight less than 89.7 tons of coal? (Round standard deviation X to two decimal places and your answer to four decimal places.)

C) What is the probability that 31 cars chosen at random will have mean weight less than 89.7 tons of coal? (Round standard deviation X to two decimal places and your answer to four decimal places.)

 

Answer 1

To find the probability in both cases, we can use the central limit theorem. The central limit theorem states that the distribution of the sample means of a sufficiently large sample drawn from any population will be approximately normally distributed. In this case, we have many cars (12 and 31), and we can assume that the sample means will be normally distributed.

The mean of the population is 90 tons (μ), and the standard deviation of the population is 3 tons (σ).

For part B, where we have 12 cars:

1. The sample mean (X̄) of 12 cars will also have a mean of 90 tons, the same as the population mean.
2. The standard deviation of the sample mean (standard error) is calculated as σ/√n, where n is the sample size. In this case, n = 12. Standard error (SE) = 3/√12 ≈ 0.866

To find the probability that the mean weight of 12 cars is less than 89.7 tons, we can standardize this value and use the Z-score formula:

Z = (X - μ) / SE Z = (89.7 - 90) / 0.866 Z ≈ -0.346

Now, we can find the probability using a standard normal distribution table or calculator. The probability that Z is less than -0.346 is approximately 0.3632.

So, the probability that 12 randomly chosen cars will have a mean weight of less than 89.7 tons is about 0.3632 (rounded to four decimal places).

For part C, where we have 31 cars, we will repeat the same steps with n = 31:

1. The sample mean (X̄) of 31 cars will also have a mean of 90 tons, the same as the population mean.
2. The standard error (SE) is calculated as σ/√n, where n = 31. SE = 3/√31 ≈ 0.537

To find the probability that the mean weight of 31 cars is less than 89.7 tons, we can standardize this value:

Z = (89.7 - 90) / 0.537 Z ≈ -0.530

Using a standard normal distribution table or calculator, the probability that Z is less than -0.530 is approximately 0.2981.

So, the probability that 31 randomly chosen cars will have a mean weight of less than 89.7 tons is about 0.2981 (rounded to four decimal places).