Boxes are labeled as containing 400 g of cereal the machine...

Boxes are labeled as containing 400 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. (a) If the target weight is 400 g, what is the probability that the machine produces a box with less than 375 g of cereal? (Round your answer to three decimal places.) (b) Suppose a law states that no more than 10% of a manufacturer's cereal boxes can contain less than the stated weight of 400 g. At what target weight should the manufacturer set its filling machine? (Round your answer to the nearest gram.)

Transcript

00:01 Okay, so we are doing a normal distribution problem.

00:06 So we're filling up some boxes.

00:08 The mean was 400, the standard deviation was 12.

00:11 I'm going to be using staplet .com, but i can tell you the calculator commands if you're using like a ti -84.

00:20 So first, oh, i guess i accidentally click on that, go to staplet .com, you're going to click on normal distributions in the probability section.

00:28 If you're on a ti -84, you're going to go to second bars and then to normal cdf.

00:48 Okay, so you need the mean and standard deviation, which was $40012.

00:56 And we're going to calculate the area to the left of $375.

01:00 So it says less than $375.

01:03 So instead of between, we'll change this to the left.

01:09 375 and calculate.

01:13 So the probability is 0 .0186.

01:20 Okay, so now they're saying, okay, we don't want to set it to 400 because as you can see, half of your boxes will be less than 400 and the other half will be over.

01:30 We don't want any more than 10 % to be under 400.

01:36 So that means we're going to move 400 over here.

01:41 We're going to to have this area be 0 .1 or 10 % and we're wanting to know what should this new mean be.

01:50 That's what we're trying to do now.

01:52 So we're working backwards.

01:54 So we're going to calculate a value corresponding to an area or if you're in your calculator you're going to do second bars 3, which is inverse norm.

02:09 Invers is when we're working backwards.

02:11 So that makes sense.

02:13 Okay, so we want to change this because now we don't know what the mean is, we need the mean to be zero and standard deviation to be one.

02:24 That's going to return a z score to us.

02:30 And we want the boundary that's going to make the left tail area be 0 .1 or 10%...

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