Coal: Automatic Loader Coal is carried from a mine in West...

Coal: Automatic Loader 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 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean $\mu=75$ tons and standard deviation $\sigma=0.8$ ton. (a) What is the probability that one car chosen at random will have less than 74.5 tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight $\bar{x}$ of less than 74.5 tons of coal? (c) Interpretation Suppose the weight of coal in one car was less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars selected at random had an average $\bar{x}$ of less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Coal: Automatic Loader 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 75 tons of coal into
each car. The actual weights of coal loaded into each car are normally distributed, with mean $\mu=75$ tons and standard deviation $\sigma=0.8$ ton.
(a) What is the probability that one car chosen at random will have less than 74.5 tons of coal?
(b) What is the probability that 20 cars chosen at random will have a mean load weight $\bar{x}$ of less than 74.5 tons of coal?
(c) Interpretation Suppose the weight of coal in one car was less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in
20 cars selected at random had an average $\bar{x}$ of less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters, the mean (?) and the standard deviation (?), and is fundamental in statistics for modeling real-world phenomena where data tend to cluster around the mean. This concept is crucial because many natural and man-made processes, under certain conditions, can be approximated by a normal distribution, allowing for predictions about probabilities and outcomes.

Standardization and Z-Score

Standardization is the process of converting a random variable to a standard normal variable by subtracting the mean and dividing by the standard deviation, resulting in a z-score. The z-score indicates how many standard deviations an observation is from the mean. This concept is important for comparing individual data points from different normal distributions and for calculating probabilities using standard normal distribution tables.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of the sample means computed from a large number of samples drawn from the same population. This distribution has a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. Understanding this concept allows for determining how sample means behave and is essential when making inferences about the population based on sample data.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size becomes larger, regardless of the underlying distribution of the population. This theorem is a cornerstone of statistics, as it justifies the use of normal distribution methods in hypothesis testing and confidence interval estimation for sample means, even when the original data do not come from a normal distribution.

Hypothesis Testing and Process Control

Hypothesis testing is a statistical procedure used to determine if there is enough evidence in a sample of data to infer that a certain condition or effect exists in the larger population. In the context of process control, such as ensuring the accuracy of a loading machine, hypothesis testing is used to decide whether observed deviations from a target value indicate normally expected variability or a potential malfunction. This concept helps in making decisions about whether to adjust or investigate a process based on statistical evidence.

Transcript

00:01 So we have a distribution that is approximately normal, which means now we don't care how big the sample size is.

00:09 And we have the mean amount in being loaded into a car, coal, i believe it was, is 75 tons, and the standard deviation was 0 .8 tons.

00:22 And we have, what is the likelihood that you pick one car and it holds less than 74 .5 tons.

00:33 And because, again, this is a normal distribution, we can answer this question.

00:37 But we're non -normal, we wouldn't be able to answer it because the sample size is one.

00:42 So we need to convert that to a z value, and we know that we're going to take that 74 .5 minus the 75, and then divide it by 0 .8.

00:52 And that z value gives us, and that is a negative 0 .5, divided by 0 .8 and that is a z value of negative 0 .625.

01:06 Now if you're going to look it up in your table, you'll probably be using negative 0 .63.

01:10 And so when i look up negative 0 .63, that corresponds with 0 .2643.

01:16 Now if you use your software with normal cd -up, you'll get a little bit different answer, but it's still going to be pretty close to that.

01:23 Then part b asks, what if you use, take a sample size of 20 and that will be sufficient to cause my sampling distribution to be normal because my original distribution was normal.

01:35 And what's the likelihood of getting an x bar that is less than 74 .5? and think about what you're doing.

01:41 You're taking 20 cars at random and then adding together and finding out if that average is less than 74 .5.

01:49 And so we want to convert that to a z value and we have that 74 .5 minus 75.

01:56 So we have the numerators...

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