for a standard normal distribution, if mc007-1.jpg, what is the value of mc007-2.jpg? This is a topic that many people are looking for. savegooglewave.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, savegooglewave.com would like to introduce to you Probability with the standard normal table (P(0 to z)). Following along are instructions in the video below:
This video. Well be looking at how to find probability using the standard normal normal table. When youre using a standard normal table that gives probabilities between a of zero and a z value and well work on these eight examples before we get started lets make sure that you are using the same z table that im going to be using.
There are multiple versions of the standard normal distribution or the z table. The one on the far left is the one well use in this video. You can see the difference between this version and the other two by taking a look at what is the first value in the upper left of the table.
Here you can see it says z at zero has a probability of zero. And you can see that the shaded area above is showing shading between 0. And a z value.
If your table shows something like shaded area to the left of z and a value of any other probability than 0 in the top left. Then youre using a different version of the table. Now lets talk about how to read this table.
The interior of the table is probabilities for example. This number right here 3686 is 3686. if i had solved a problem that gave me a z value of 1121437.
The z table here gives us zs that go out to two decimals. The first column of the table is the first two digits of the z and the first row above is the third digit of the. Z so if we had a z of 1121437.
We would be able to find probability in this table for a z of 112. Heres how you find a z. We look at the first column and we find the first two digits and then we go to the first row and we get the third digit of 002.
The intersection of these two values tells us the probability. And the probability that we have just found is between the mean and a z of 112. 3686.
. When we talk about this we talk about area under the curve or. Probability.
For. Z at 112. In this table.
It gave us a probability of 3686 . If we look at one of the other tables for z for example in this. Table if i looked at 112.
You can see that the value is different. Its 8686 because this table is giving me the probability. Of z less than.
112 and that is 08686. Which is why its important to make sure youre using the same table. When you look up probabilities as the person in the video.
Now lets get started working on some problems in this first problem. Were going to find the probability between 0. And 1 and how we write this is the probability of 0.
Less than z less than 1 lets begin by drawing out a normal curve and showing. 0. Which is the mean on the standard normal curve 1 2.
3. And these represent number of standard deviations from the mean. And i want to find the area that is between the mean and z of 1.
This piece right here well it turns out when i look this up in the table. Im going to be looking up 1 in the table. And when i do im actually finding this first value right here.
The answer to my question. This is what i want to know the probability between the mean and one. And that is the value given in the table.
Now lets look at one. Where we have to do a little bit of work to get to the answer in this one. Were going to find the probability of z.
Greater than 1. Lets draw this out and here is one on this table. So this would be z equals.
One and we want to find a probability greater than one this is what we want to find the probability of z. Greater than one when we look this up in the table. However what we look up in the table will be the probability thats between 0.
And 1. This is what the table will give us one property of the standard normal table that you must understand in order to solve these kinds of problems is that 50 of area is to the right of the mean the probability of z greater. Than 0 is.
50 . If i want to know the probability of z. Greater than 1 and i can subtract out from my 50.
Some piece thats up to 1 the leftover will be the chunk that i care about right here. This is going to be the probability of. Z greater than.
0. Minus the probability of z between 1 and. 0.
050 or. 50 minus the probability. Between 0.
And 1 03413 and the final answer that i get. Is. 01587 or 1587 of area is to the right of z.
At one which we say as the probability of. Z greater than 1. Equals.
01587. Now lets take a look at another and this one will be solving the probability. Of z being.
Between 108 and. 245 lets find those values in. The table.
First 108 is going to be at this intersecting point. Of 03599 and. 245 is going to be at this.
Intersecting point of. 04929. Lets draw this out on the standard.
Normal curve i want to find. The area thats between. 108 and 245.
Right here this is what we want to find what ive looked up for. 108 however is the area that is in between z. Of 0 and.
108 and the area between z of. 0 and 245. And in order to get this piece right here.
I will need to do a little bit of math. What im going to do is im going to take the larger percentage ive found which is the area thats between the mean and 245. And im going to subtract from it the area between the mean and 108.
And the leftover is going to be the result i wanted to get which will be the area. Between 108. And 245 so im going to take 04929 and subtract.
From it 03599 to get my. Answer. Of.
015. 30 153 of area is between. 108 and 245.
Standard deviations. Now lets take a look at if we want to find the probability. Of z less than.
167. Again well draw the curve and 167. Is right about here and i would like to find everything to the left of this however when i look this up on the.
Z table its going to. Show me. The probability at.
167. 04525. So ive just found this bit 4525.
We know that everything to the left of zero is 50 of area. So if i know the piece thats to the right of zero that i care about and i actually know what area is to the left of zero in order to find the probability of. Z less than 167.
I take the probability of z. Less than zero and add to it what i just found which is the probability between the mean and. Z of 167 so 05.
Plus 04525 gives me the. Answer. 09525.
Now lets take a look at some negative zs youll notice that this table. Only has positive z values on it this is what we call half of a table it starts at z of 0. And then gives the probability to the right of that between the mean and any value however we can use the same table for negative zs because the table is symmetric lets take a look if i want to find the probability between minus 1 and 0.
Then thats this area right here i can ignore the negative sign knowing that the same area is going to be between a z of 0. And a z of 1 as between a z of 0. And minus 1.
So i can look this up in the table. And it will be exactly the same on the left hand side of 0. Because they are mirror images of each other and we already found that answer to be 03413 just as 3413.
Is the area under the curve between 0 1. It is also the area under the curve between 0. And minus 1.
Lets take a look at another for this. One well do the probability. Of z less than minus.
196 drawing out the. Curve we want 196. Which is right about here now when we look this up in the table.
Just as it would on the right hand. Side it is going to give us the area between minus 196. And 0.
Because the table is always giving us the area between the mean at 0. And a z value the area right here. We would actually like to find this area.
Thats what weve been asked to find the probability of. Z less than minus. 196.
50 of area under the curve is to the left of 0 the probability of z. Less than 0. Is 50.
i find that at 196. I have 47 and a half percent of area under. The curve so the probability between minus.
196 and zero is. 04750 and if i subtract it from fifty percent then i get the. Answer.
00250 two and a half percent of area is to the left of minus 196 now lets look up. The area thats between 275 and. 05 well draw out.
The curve we are looking for here at. 275 and here 05. Now if we look at these values in.
The table we would ignore the minus sign and go find 050. And 275 these two values of 01915. And 04970 are the area under the curve between.
05 and the mean and. 275 and the mean in order to get the chunk that is between 05 and. 275 i need to subtract the smaller probability from the.
Larger 04970 which represents the entire distance between the mean and 275 and. 01915 which represents only that small chunk between 05. And the mean the leftover will be the probability of finding a value thats.
Between 05. And 275 five standard deviations and the answer. To.
That is 03055 or 3055 of area is between. 05 and. 275 standard deviations on.
This one well do the probability. Between 218 and. 196 lets circle those numbers in.
The table. 218 will be over here and 196. Will be here.
We have just circled numbers that represent area here probability. Of 0 less than z less than. 196 and here the probability between 218.
And the mean well it turns out that thats exactly what we want in this problem we actually want to know the probability that is between. 218 and 196 if we add these two probabilities together we will have the. Entire probability 04750 and 04854 will give us the answer of 9604 or 9604 of area is between.
218 and 196. Standard deviations in this video. We have looked at how to use the standard normal table.
Which gives half of the area under the curve from z. At zero to the right in order to solve any probability for z negative or positive. I hope you found this helpful .
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