Problem

In Exercises $32-34,$ assume a normal distributio…

04:41

Problem 32 Hard Difficulty

In Exercises $32-34,$ assume a normal distribution.
Machine Accuracy A machine that fills quart bottles with apple juice averages 32.8 oz per bottle, with a standard deviation of 1.1 oz. What are the probabilities the amount of juice in a bottle is as follows?
a. Less than 1 $\mathrm{qt}$
b. At least 1 oz more than 1 qt

Answer

(a) 0.2327$\qquad$ (b) 0.4286

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Calculus 1 / AB

Finite Mathematics and Calculus with Applications

Chapter 18

Probability and Calculus

Section 3

Special Probability Density Functions

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Video Transcript

Okay, So for this question, we're told that we have a normal random variable which will call X that is normally distributed with an average of 32.8 and a standard deviation of 1.1. So how we typically write a normal random variable his ex in this little till day and then n an imprint sees the first very first input is the mean. The expectation on second input is the standard deviation or the variance. I prefer to write as variants. So I just write 1.1 squared here. So then we want to know the probability that X is less than one court. Well, our variable is in. Let's let me quick check the conversion. What's two ounces? It's 32. Look most a little bit awkward, so we're just asking the Probability X is less than one court or 32 ounces less than or equal to 32. So how we're gonna do this is we're gonna standardize this variable X since we know it's a normal random variable, and we have all these nice tables for normal variables with a mean of one in standard deviation or a mean of zero In a standard deviation of one, we're going to standardize the sex. So how we do that is we subtract It's mean and divide by the standard deviation and what we do to one side of the inequality we have to do to both, Like so and said this thing when we have a normal variable minus its mean divided by its standard deviation. No, we get the standard normal variable, Stan. Normal distribution like that. So well, denote that with the sea. Then over here, we're gonna get ah value by playing all this in its negative 0.7272 And so now this question is something we can answer the probably that a normal random variable is less than negative. 0.7272 So we come over here to our table, we find negative 0.72 here. And so it looks like that is going to be 0.2 three. Oh, actually, I guess we should do point negative. 00.0.73 eso this value. Um, because this will round up says 0.23 27 as our first answer. Then for the next part, we want at least one ounce more than one court. So probability that X greater than 33. So doing exactly what we did over here, we're gonna get properly that see, it's greater than 33 minus 32.8, divided by 1.1. That's 0.1818 And so, if we go towards the table, we find 0.1818 It looks like we're going to get oh ah! Then something important to note. So we're looking at the probability that Z is greater than 0.1818 So that would be one minus. The probability that's he is less than or equal to 0.1818 Because that's how busy table values only give us a probability less done. This is gonna be one minus 10.57 14 which equals point for 286 So that will be our second answer

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