for a standard normal distribution, what is the probability that z is greater than 1.75? This is a topic that many people are looking for. bluevelvetrestaurant.com is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, bluevelvetrestaurant.com would like to introduce to you Stats: Finding Probability Using a Normal Distribution Table. Following along are instructions in the video below:
This video. I want to show you how to find using a standard normal distribution distribution table. How to find probabilities that are of interest for example.
If you are to find the probability that z. And well talk about z. In just a second is less than negative one point thirty two using a table.
How can we actually find that value well a probability. I want you to think a probability this thing right here is simply an area alright. Its an its an amount of area.
And i showed in a previous video that the total area under a standard normal distribution. Bell shaped curve is equal to one so we better get a value and these values here whatever we come up with will always be less than one all right let me just show you what i mean by using a table. I have various statistics books.
Let me just pick this my kalla camera up here. I have a whole bunch and so it depends on what kind of book youre using here. But i hope you can see this is one example of a standard normal table.
I cant zoom into the nitty gritty there. But you know this is one example. But i want you to pay attention to pay special attention to the picture at the top of these of these pictures of these uh tables here so it depends on what book youre using.
But pay really close attention to what the picture is telling you them and get rid of that one here heres another example of a book got this from a different stats book. And you can see this just coming from the the inside back cover of the of the book. And again.
Im just kind of paying attention especially to the to the picture. All right because the picture tells you whats going on inside. This table.
Here. Heres another example clear out all this junk. Another example got this out of a different book.
The book that i actually use quite a bit in my classes is a trio thats name of the author called elementary stats. And this is the table that he uses so im paying really close attention to the picture up at the top of these tables all right and notice at least in this table that im using that theyre shading going on and the shading is to the left okay. So what im going to do here to go back to my problem is this problem that im i posted here that i just put is assuming that were looking to the left thats a less than.
Symbol so were looking to the left of. Negative 130. To picture wise picture wise.
Heres whats going on. I have a bell shaped curve because its standard you can look this up in a previous video that means that the mean is sitting right in the middle. Right.
Here ddddd and so that means that everything to the right hand side of 0. Is positive right everything over here is positive and everything to the left hand side of 0 is negative well that makes sense on the number line. I think all this stuff is negative over here ok so if i was to place if i was to try to find a negative 1 point 32.
I hope you see that thats got to be somewhere on the left hand side of 0. So its got to be something over here it doesnt really matter where you put it im going to stick it over here somewhere so ive got a negative 1 point 32. That is the z score that i have right now and since that is a less than symbol that means were shading everything to the left of it im shading to the left now.
According to the table that im using right. According to the table that im using. Which is this book right here im going to use the negative z.
Score tables there it is right there im going to use this thing this negative z. Score table because it shades to the left thats exactly what im after so im going to scroll down here and look for a negative 1 point 32 and anyways if you can look it up in your own book. But you can see that hopefully youll get a value of point zero nine three four point zero.
9 3. 4. So theres my answer for this first problem that im going to present ok.
So point zero three four typically youre using mymathlab especially it wants it rounded off to four decimal places. And thank goodness the table already does that for you so thats what thats the picture. Thats going on maybe.
The picture is really helpful to you visually to see and because that was a less than symbol. I shaded to the left and thats what my table actually gave me so i just looked up a negative one point thirty two in the table. And i came up with that that.
Probability so what im trying to say is this number 009. 34. Corresponds to this area right here.
All of this area.
Right. Here. Underneath the curve that now a shade in blue is 009.
34. Which is definitely less than 1. Right.
The total area under the curve is 1. And that is definitely not the total area. So.
Its point. O 934 or some people look at that. Decimal and say.
Hey thats about 9. Percent. Right 934.
Percent. To be exact okay lets change. The problem up a little bit.
What if they asked all right what if theyre asking for something like this whats the probability that z is greater than negative 1 point 32. All right so now this time were not looking for less than were not going to shade to the left this time. Were going to shade to the right.
So this is what im looking for lets draw a picture because again the picture really is helpful zero sitting right in the middle and you can see that negative one point thirty two is somewhere over here to the left hand side of 0. But im not shading to the left this time this time. Im shading to the right everything im looking for all of this area over here.
And why because this is a greater than symbol right greater than greater than and greater than means to the right okay. So i cant just use the table that i presented with over here. I cant just use this table as is because thats a shading to the left.
So i need all of this area over here to the right of that curve well if i use the table as you saw in the previous. Example i just did i had an area of 009. 3.
4. Right in other words all of this stuff over here right. All this area.
Over. Here that i just shaded in blue is 009. 3.
4. Well. Because you know that the total area is one then this area.
Over here in red must be its complement complement. Means 1 minus. 009.
3. 4. So im going to work this out.
And thats all of this red area over here well i can just do that very easily on my calculator or you can probably figure it out doing it. Mathematically no problem. Turn this guy on and punch in my 1 minus.
009 3. 4 and youll see that ive got a value of. 09.
O66. Okay so this total area. Over here is.
09. O66. Which think about that for a second folks think about that picture wise for a second if all of this area is 1 and just this little much over here that on the left hand side of negative 1.
Point 32 is 009. 34. Then the majority of the area under the curve sitting on the right.
Side which is all this red.
Stuff that i highlighted is. 009. Oh.
6. 6. Yeah.
Thats roughly 90 or 91. Percent of the total data. Okay one more last example is this one what if sometimes they want you to shade in between right.
So sometimes you get a problem where they say look find the probability that we have some z number. And were looking for the stuff thats in between how about negative zero. Point.
21 and im just making these up here 085. This time i want the area thats in between those two z numbers. All right well lets go to the picture.
You know that phrase a picture says a thousand words or something like that all right so here is our picture zero sitting right in the middle where is a. Positive 005. Well thats got to be to the right of zero.
Because all these numbers over here are positive so point 85 is sitting over here. And where is negative point. 21.
Oh. Thats got to be on the left side of 0. Because thats where all the negatives sit point 21.
So were looking for hope you see and i wasnt being very exact of where i stuck these things were they just i know that point 85 is on the right and i know that negative point. 21. Is on the left.
It doesnt really matter where. But as long as theyre on the right hand the correct side. I should say look i want all of this area thats in between them all right all the area in between them so it again it depends on what kind of table youre using.
But when i use this table here that comes with the book that im using and its shading to the left what im going to do is im going to look up two things im going to look up point 85. And im going to look this up point. 85 which zero this is a positive number so 085.
I hope you might agree with me comes out to be all right this one over here comes out to be 08. Oh. Two three all right again that came from looking this up on the table.
Im also going to look up negative zero point twenty one negative zero point twenty one okay thats not too bad thats 41 68. 168. And so what i actually looked up folks.
I hope you see are these areas. I looked up this 802 three actually represents all of this area right here that ill shade in green for you okay. It actually represents all of this area here okay.
Thats what the 802. 3. Represents and then i looked up this negative point.
21. That gave me an area of 40 168. Thats actually this area shade over here in black alright.
So what im doing is im going to take all of this green stuff take away this black stuff here and im left with what i shaded originally. Which was in red thats a geometry problem really right so im just going to take the difference of these two numbers. Im going to subtract them.
And i hope you see that what we are left with is right what were left with is the total area in between those two z numbers. So here it is point 4 six eight. Im just kind of doing this on my calculator.
But ill show you what i just did okay i punched in point. Eight oh two three minus. 0416.
Eight. Im going to hit equals and ive got 38 55. That is my total area in between point 38 55.
There we go four numbers rounded to four decimal places. Thats what i would punch in on mymathlab that is this area sitting right here in the middle right in the middle point 38 55. .
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