A manufacturing plant produces metal rods for automotive assembly. Based on quality control data, rod lengths are normally distributed with μ = 100 cm \mu=100\operatorname{cm} & σ = 0.8 cm \sigma=0.8\operatorname{cm} . Rods shorter than 98.5 cm 98.5\operatorname{cm} are considered defective. What % \% of rods are below this tolerance?

Folks. So now that we understand how to find probabilities using the nonstandard normal distribution, let's take a look at some practice problems and see how this works in the real world. So we have a manufacturing plant that produces some metal rods for use in automotive parts. And based on quality control data, rod lengths are normally distributed approximately with a mu of a 100 and a standard deviation of eight. In other words, we're still working with a sort of bell curve like distribution over here, except instead of our mu being zero and sigma being one, we just have different numbers. But it works the exact same way. In our case, the mu is a 100. That's where the center of this normal distribution is. And to measure how spread out the data isn't one, it's just the sigma here, which is 0.8. And we're supposed to find is rods shorter than 98.5, which is a sort of cutoff or threshold are considered to be defective. So what percent of rods are below this sort of tolerance or within this sort of threshold of being defective. Alright? So if our mu is equal to a 100 over here, then 98.5 would be to the left of that. In other words, what I'm trying to find over here is there's this sort of number I can go ahead and just I'm just gonna draw a line over here somewhere. This is gonna be where 98.5 is. That's gonna be my x value. Anything that is below this sort of cutoff or threshold is gonna be considered defective. And basically, what I'm trying to find here is what is this percent. In other words, what I'm trying to find is what is the area under that curve over there, right, or the defective rods. Alright. So in other words, I have my mu, which is equal to 100. I've got my sigma, which is 0.8. I've got my x value, that specific x value that I'm interested in, which is 98.5. And what I'm ultimately trying to do is I'm trying to find a probability. What's the probability that a randomly selected rod that I find is below that 98.5? Now this is a nonstandard normal model, but I can always take these numbers and I can transform them back into the standard model by using our new z score equation. So let's go ahead and use that over here. So I'm going to use the z score equation, which is x minus mu over sigma, which is basically just if I go ahead and start plugging in some numbers, this is going to be 98.5 minus 100 divided by 0.8. What you should find here is that z score is equal to negative 1.875. You can go ahead and just round it up to two decimal places, so I'm just going to go ahead and say that this is negative 1.88. In other words, in this model over here, an x value of 98.5 really just translates to a z score of negative 1.88. Alright? So in other words, the probability of finding a raw that is less than 98.5 is the exact same thing as saying, what is the probability that in the standard normal model, that I would find something with a z score of less than negative 1.88? It's the exact same problem. So you can go ahead and consult with a z table like the one that we have over here, or you can go ahead and use a calculator. But what we would do is we'd go into the negative numbers over here. We would go up to 1.8, and then with a decimal of in the hundredth place of eight, we would find this is about 0.0301. Right? So in other words, this is really just going to be a probability. So probability that x is less than 98.5 is just gonna be equal to 0.0301, which is about 3.01%. Alright? That's about how how many of these rods would be considered defective. Alright? That's it for this one, folks. Let me know if you have any questions, and let's move on.

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Multiple Choice

A manufacturing plant produces metal rods for automotive assembly. Based on quality control data, rod lengths are normally distributed with $mu equals 100 cm$ & $sigma equals 0.8 cm$ . Rods shorter than $98.5 cm$ are considered defective. What $%$ of rods are below this tolerance?

$1.24 percent sign$

$2 percent sign$

$3 percent sign$

$5 percent sign$

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검증된 단계별 안내

Step 1: Understand the problem. The rod lengths are normally distributed with a mean (μ) of 100 cm and a standard deviation (σ) of 0.8 cm. We need to calculate the percentage of rods that are shorter than 98.5 cm, which means finding the cumulative probability for this value in the normal distribution.

Step 2: Standardize the value of 98.5 cm to a z-score using the formula: z = (X - μ) / σ. Here, X is the value of interest (98.5 cm), μ is the mean (100 cm), and σ is the standard deviation (0.8 cm).

Step 3: Substitute the given values into the z-score formula. This will give you the z-score corresponding to 98.5 cm.

Step 4: Use the z-score obtained in Step 3 to find the cumulative probability from the standard normal distribution table or a statistical software. This cumulative probability represents the proportion of rods shorter than 98.5 cm.

Step 5: Convert the cumulative probability into a percentage by multiplying it by 100. This percentage is the answer to the problem, representing the proportion of rods below the tolerance level.