Find the z-score such that P P P (Z < z) = 0.6331 =0.6331 = 0.6331 .

Everyone. So welcome back here. Let's take a look at this practice problem. Again, now we're dealing with a slightly different problem here. We're gonna not find a probability, but find a z score such that the probability of all the values to the left or less than some unknown z value, which we're gonna write here, which we're gonna find, is equal to this number, 0.6331. Remember, these types of problems, when they're written this way, this number over here is not a z score. It's a probability. Okay? Now, again, before you go off to your z table and start looking up some numbers, always a good idea to just actually draw out the sketch of the graph here. Okay? Now in most cases, what happens in these problems is you maybe have already given the graph, but not in this one, and so we're gonna have to go ahead and sketch it. So before, we actually knew where to draw our vertical line to shade our area. We actually don't know in this case, and all we have is just the area over here. Here. So how can I use this to figure out where the line's gonna be? Well, basically, I just look at this number here, and if it's close to zero, like it's like point zero one or something like that, I know the line is gonna be over here way off the left because all of this area here is gonna represent a tiny little piece of the whole entire thing. It's gonna be a very small number close to zero. If it were close to one, like point nine or point eight or something like that, the line would be somewhere over here because then all of this area would be close to one. Right? But this is somewhere in the middle. It's like point six. Right? If the line were perfectly in the middle, then this would just represent half of the normal curve, and that would actually be exactly point five. This is slightly more than that. So in other words, I'm just gonna draw the line a little bit to the right of that middle line over here such that the shaded area that I'm gonna draw in yellow is actually a little bit more than half, like, that that's 0.63. Okay? So this is actually the area that we're interested in. Okay? So I know the z score is gonna be somewhere over here. And, again, this is just gonna be a rough sketch. Alright? So this is our z score. So remember how to do these types of problems here? You're gonna have a probability that's given to you by the problem, and that and that's actually this number right over here. So it's 0.6331. But that's not always the probability that you're gonna go to your z table and look up. So we're gonna have to just figure out, do we actually have to do anything to this probability first before we go look it up in the table, or can we just look it up as it is? And again, it all just comes down to which area you've shaded, right? In this case, what happens is we've actually already indicated that it's going to be the area to the left of that z score, which is exactly the values that you'll see in your z table. So, in other words, you actually don't have to do anything. So, probability given is 0.6331, and the probability to look up is also equal to point six three three one. Alright? So let's go over to our z tables over here and try to find this value. So basically what happens is as you go down the row on the left, the numbers will get bigger, And then same thing over here, on the right side, except they'll just start from point five. So we already know that we're gonna be dealing with a positive z score somewhere over here. And, again, I try to just just look for all the point sixes. Right? I'm looking for as a a probability of an area of 0.6331. So let's go ahead and go through our values. I've got 0.6, I've got 0.63, and it looks like it's right over here. So I found my 0.6331. It's this number right over here. And remember to find the z score, I basically had to do the opposites. I just have to go ahead and draw the rectangle that includes that and then draw the column. And then what I'm actually looking at over here is I'm going to look at these numbers. So I'm going to look at 0.3, 0 0 4, and then basically just reconstruct the z score. Right? So in other words, this area that corresponds to the z score over here is gonna be 0.34. Alright? So therefore, the z score that we're interested in is 0.34. And by the way, that perfectly makes sense relative to where we've drawn it. We knew there was gonna be a number that's kinda close to zero, but it's gonna be a decimal, and it's gonna be a little bit closer to zero than it is to one. Right? Alright. So that is our z score, and that's basically our answer. Alright? So that's it for this one, folks. Let me know if you if you, have any questions, and let's move on.

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Multiple Choice

Find the z-score such that $P$ (Z < z) $=0.6331$ .

0.34

0.75

0.62

0.66

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Verified step by step guidance

Understand that the problem is asking for a z-score corresponding to a cumulative probability of 0.6331.

Recall that the z-score is a measure of how many standard deviations an element is from the mean in a standard normal distribution.

Use a standard normal distribution table (z-table) or a statistical software to find the z-score that corresponds to a cumulative probability of 0.6331.

Look up the value 0.6331 in the body of the z-table to find the closest probability, and then identify the corresponding z-score from the margins of the table.

Verify the z-score by checking if the cumulative probability for the found z-score matches 0.6331 using the z-table or software.