A company is launching a new line of smart thermostats & \& predicts that weekly sales will follow a normal distribution with μ = 4 , 000 \mu=4,000 units & \& σ = 500 \sigma=500 units. How many units must be stocked each week to ensure that demand is met 95 % 95\% of the time?

Now that we understand how to find values from given probabilities in a nonstandard normal distribution, let's take a look at our practice problem and see how this works in the real world. Hopefully, you had a chance to work it out in your own. So we have a company that is launching a new line of smart thermostats, and they predict that weekly sales is gonna follow a normal distribution. In other words, it's gonna be shaped kind of like a bell curve. Now instead of our mu being zero and sigma being one, like we've normally dealt with this, we're gonna have that mu is equal to 4,000 and sigma is equal to 500. I always like to start with a little picture here, so I'm going to go ahead and just draw a sort of bell curve or a normal distribution. We have that mu is equal to 4,000, and our sigma, which is kind of just a measure of how wide our distribution is, is 500. And we're supposed to find how many units must be stocked each week to ensure that demand is met 95% of the time. So what does that mean? Basically, we have a warehouse with a bunch of thermostats in it, and we're gonna sell on average from week to week around 4,000 units. Sometimes it'll be less, and sometimes it'll be more than that. We need to have enough in our warehouse stocked to ensure that we can meet that demand 95% of the time. Okay? So in other words, we're not looking for a probability or an area here. We're actually looking for an amount of units. So in other words, what's going to happen here is there's going to be a number that's going to be above 4,000 such that, basically, you can fill that demand 95% of the time. In other words, all of the area to the left, basically, all the units that we have in the warehouse, is gonna be enough to satisfy that demand 95% of the time. So this is gonna be our area to the left over here, and it's gonna be 95%. Or as a decimal, this would be 0.95. This is what we're actually trying to find here. What is this x value such that that area to the left is 95%? So we're gonna go ahead and use our, our new equation here, which is kind of just a rewrite of the z score equation to figure out what that x value is. So I've got that mu is equal to 4,000. We've got that sigma is equal to 500. And I've got the area to the left is gonna equal 0.95. Alright? Now before you go ahead and just start working out this, equation over here, let's take a look. We have the x is equal to z times the sigma plus our mu. Now we actually already have what sigma and mu are. Right? So we have sigma and mu. The only thing is we don't know what that z score is. However, we can find that z score because we do have what the area to the left of this value is, and we know that it's gonna be a normal distribution. So what we do is we go from area to z score basically, just by using either our z table or some technology here. We're going to have to figure out what is the z score that translates to 95% of the area being covered to the left. So this is where you'd get out your z table, or you can go ahead and just use your technology for this. Basically, what you're going to do is you're going to come over here to the z table. And in this, what you're actually doing is you're sort of doing the opposite of what we've normally done with looking up z scores. We're actually going to look up the area that you have to the right to the left, and you're gonna figure out the corresponding z score. We've done this before. So I'm gonna go ahead and look for where 0.95 is. If you go ahead and look at these values over here, you're gonna get pretty close, and you're gonna get somewhere over here. So what I'm actually looking at is I'm looking at these two values, 0.94950.9505. Somewhere in the middle there is 0.95 exactly. So what I'm going to do is I'm going to go all the way to the ends of the rows and columns over here, and it's just going to be these values. So what you think you could do here is you can kind of just take the average of one point six zero four and zero five, and you can just go ahead remember, that's the hundredth place there. And you should recognize this value when it comes to that z score, which is gonna be 1.645. Alright? This is a number that should be pretty familiar to you. We've used it a bunch of times, and that's the that's the z score that corresponds to 95% of the error being on the left. Okay? So remember, we need that because we need to plug it into this equation over here. So in other words, to figure out our x, we're just going to take that z score that we just found at 1.645 and we're going to multiply it by our sigma, which in this case is 500, and then we're going to add it to the mu, which is 4,000. What you should get over here is you should get about 4,822.5. That was your answer. Now because we're dealing with, you know, specific amounts of a discrete unit, like a thermostat, I'm just gonna go ahead and round up to the next highest number here. In other words, we need to have 4,823 of those thermostats in order to satisfy demand 95% of the time. Alright? So this is, again, how we would use sort of this z score equation to figure out a value from a given probability. Let's go ahead and take a look at our next problem. Thanks for watching.

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Multiple Choice

A company is launching a new line of smart thermostats $&$ predicts that weekly sales will follow a normal distribution with $mu equals 4 comma 000$ units $&$ $sigma equals 500$ units. How many units must be stocked each week to ensure that demand is met $95 percent sign$ of the time?

$4600$ units

$4900$ units

$5000$ units

$4823$ units

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Identify the problem: The goal is to determine the number of units to stock each week to meet demand 95% of the time. This involves finding a value in a normal distribution corresponding to a cumulative probability of 95%.

Understand the given parameters: The problem specifies that the weekly sales follow a normal distribution with a mean (μ) of 4,000 units and a standard deviation (σ) of 500 units.

Use the Z-score formula to find the value corresponding to the 95% cumulative probability. The Z-score formula is Z = (X - μ) / σ, where X is the value we are solving for. Rearrange the formula to solve for X: X = Z * σ + μ.

Determine the Z-score for 95% cumulative probability. Using a Z-table or statistical software, find the Z-score corresponding to 0.95 cumulative probability. This value is approximately Z = 1.645 for a one-tailed test.

Substitute the Z-score, mean (μ), and standard deviation (σ) into the formula X = Z * σ + μ. This will give the number of units to stock each week to meet demand 95% of the time.