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Standard Normal Table Review

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Source: NORMAL DISTRIBUTION GRAPH CREATED FROM PUBLIC DOMAIN HTTP://COMMONS.WIKIMEDIA.ORG/WIKI/FILE:Z_CUMULATIVE.SVG Z-Table created by J. Gearin

Hi. This tutorial serves as a standard normal table review. So let's start with some motivation. The distribution of lengths of human pregnancies is approximately normal. That's pretty important there, the approximately normal. With a mean of mu equals 270 days and a standard deviation of 15 days, notice these are both population parameters. So the question that we might want to address is, what is the probability of a randomly selected woman having a pregnancy that lasts 252 days or fewer? So hopefully, by the end of this tutorial, we will be able to answer this question and questions like this question.

So to find probabilities involving normal distributions, we need to do two things. Number 1, calculate the z-score for a given population value, and then use the z-score and a standard normal table-- otherwise known as a z-table-- to look up the needed probability. So define your standard normal table or your z-table. It's a table that gives cumulative normal curve areas. And you think of that area under a normal curve as a probability for various values of z. So standard normal tables are frequently used in inference procedures.

Just so you get a look at what a standard normal table looks like, this is one example of a standard normal table. So you'll see that up here, it gives you your normal curve. You have a z-value here. And then notice this gives you a shaded area. So that shaded area represents the probability lying below an indicated value of z.

So now, to read the table, notice we have values of z here. And we have another value of z here. So in this table, I have z values ranging from negative 3.4 all the way to 0. And then there's a second piece of the table that will give me positive z-scores-- so z-scores here, ranging from 0 to 3.4.

Now, the values along the top of the standard normal table represent the hundreds place. So here you have numbers in the ones place and the tenths place. Here I have numbers in the hundredths place. And then all of these numbers here represent probabilities to the left of the indicated z-score.

So let's actually calculate some probabilities to get a little bit of fluency in using your z-tables. So let's just go back and try to answer this question. So what is the probability of a randomly selected woman having a pregnancy that lasts 252 days or fewer? So what we're looking for here is the probability that x is less than or equal to 252.

So this is kind of the goal, it's what we're trying to figure out. And what I think is helpful is to start by actually drawing that normal distribution. I'm going to draw it. This represents my x-axis. I'm also going to put a z-axis down here.

So the x-axis is going to be-- or the distribution on the x-axis is going to be centered at the mean, which is 270. And that represents a z-score of 0. So 0 standard deviations from the mean, so z-score of 0.

Now, over here at 1 standard deviation, this is going to end up being 285. So it would be the mean plus a standard deviation of 15. That's going to be at a z-score of 1. And then if I go down one standard deviation, if I go 15 less than 270, I'm going to be at 255. That's going to be a z-score of negative 1.

And then let's also do two standard deviations above the mean and then two standard deviations below the mean. Two standard deviations above the mean is going to be 300. And two standard deviations below the mean will end up being 240, z-score of negative 2 here, positive 2 here.

So now I want to place 252 on my distribution. So 252 is going to be a little bit to the left of 255. And I want less than. So I'm really trying to figure out what that area is.

So if we think about that, we, first of all, need a z-score for 252. So remember our z-score is going to be x minus mu over sigma. So for 252, it's going to be 252 minus 270 divided by 15. And if we end up calculating that, we're going to get a z-score of negative 1.2.

So my goal now is to figure out the probability of z being less than negative 1.2. So let's now go to the z-table. And it's a negative z-score. So I'm going to use the part that just uses the negative values. And I want the area to the left, or the probability to the left of negative 1.2.

So what I'm going to do is look on this column. I'm going to find negative 1.2. And then the hundredths value is 0. So I'm just going to look in this first column. And just see you can see it, I'm going to be looking for this value right here, so 0.1151.

So if we then-- I'm going to put that in to what we had before. The probability ends up being 0.1151. And so then to-- let's put a less than or equal to sign there-- to answer that original question, there is a about 0.1151 probability of a pregnancy lasting 252 days or fewer. So about 11 and 1/2% chance that a woman's going to have a pregnancy of 252 or less.

Let's try another one. So what is the probability of a randomly selected woman having a pregnancy that lasts 275 days or more? So now what we're looking for is this probability-- so x being greater than or equal to 275. So if we go back to this original distribution, now, if I wanted 275 or more, I'd be marking it at about here. And then I'd be shading now to the right. So we're going to have to deal with this a little different, because now I'm looking for a right probability instead of a left probability.

But let's still just calculate a z-score. So z equals 275 minus 270 over 15. And then this will end up being 5 on the top, 15 on the bottom. So that would reduce to about 1/3. So I'm going to estimate this to be about 0.33. I'm going to round it to the nearest hundredth, because that's what the z-table will show.

So let's consider this-- so we're looking for now the probability of z is greater than or equal to 0.33. So again, let's go to our z-table. Now, our z-table again shows probabilities to the left. So what I want to do-- so I'm on the negative z-table here. But remember, I wanted a positive z-score. So I'm going to switch to the positive z's. And in this case, I want 0.33. So I'm going to look for 0.3 here. And then I'm going to move over to 0.03 up here. So 0.33 is going to be 0.6293-- so 0.6293.

So let's make sure we understand what this value tells us. What that value tells us is the probability of z being less than or equal to 0.33 is that. Now, we want the probability that z is greater than or equal to 0.33. So what this probability is, and the probability that we're looking for, those are complementary probabilities. So what I can do is I can take 1 minus this probability to get the greater than probability.

And then if I end up subtracting those values, what I'm going to end up with is 0.3707. So the probability of a pregnancy lasting 275 days or more ends up being about 0.3707, or so about a 37% chance of a pregnancy lasting 275 days or more. So we had to use that complement rule to determine this probability.

Well, that has been your standard normal table review. Thanks for watching.