00:01
Suppose you are working with a data set that is normally distributed with a mean of 250 and a standard division of 48.
00:07
Determine the x value from the following information round your answers and the z values to the nearest two decimal place.
00:15
So the first question says 70 % of the values are greater than x.
00:20
So we have our m to be close to 250 and then we have our standard deviation x, a zigma rather, to be caused to to 48 if also standardize the value of x which is actually unknown at this moment we have the formula that says z is equal to x minus mou divided by z so our first question says that we should get the value of x for a 70 percent of the values are greater than x so when we standardize the value of x in this case of us we have 70 percent of the value will be greater than z so this is going to be uh excuse me this is going to be 70 percent so the z and we have this to be 70%.
01:09
So the area towards the right is greater than z and that has 70%.
01:14
And this region right here is going to be 30%.
01:18
So 30 % towards the left or 70 % towards the right is going to correspond to a z value.
01:25
So i'll be using a standard normal distribution table to get this z value.
01:30
So the table i have is actually a table towards the left.
01:33
So i'll be using this.
01:34
Area the area shaded in black so i'm going to be looking for 0 .30 on my normal distribution table so let's open our table so let's look for 0 .30 so 0 .30 should be somewhere around here now 0 .30 is supposed to be between this value this value and this value and if we trace it back we have to be between minus 0 .5 and if you trace this up here we have this to be 0 .0 .0 we have this to be and 0 .03 so we have a z score so we so i will remove the negative sign just to avoid confusion that 0 .5 plus we have 0 .02 plus 0 .03 then divided by 2 so we have ziz equals to 0 .5 plus 0 .05 divided by 2 so that ziz across to 0 .5 plus 0 .025 and that gives us zz across to 0 .525 which you can write as ziz across to minus 0 .5 to 5.
02:42
So the corresponding z square is minus 0 .5 to 5 to 5.
02:45
I recall that z is a cost to x minus mil divided by sigma.
02:51
So our z in this case is minus 0 .525 is because to our x is unknown minus our mil is 250 divided by zigma our zigma is 48.
03:00
So minus 0 .5 to 525 times 48 and that gives us a minus 10 to 5 .2 is across to x minus 250 so x equals to minus 25 plus 250 and that gives us to 24 .8 so for our first question question one the value of x is actually 224 .8 so let's go to the second question so for our second question we have a 22 percent of the values are less than x 22 percent of the values are actually less than x so let's try to have a sketch of that 22 % of the values are less than x so that has our 22 % of the values are less than x so that's this region over here 22 % of the values and this is the region of 22 which is a 0 .22 all we have to do is to go to our normal distribution table and locate 0 .2 so let's open a table 0 .22 should be somewhere around this value and this value and let's trace that that is a minor 0 .7 and if you trace it up we have 0 .708 so this simply implant our z is equals to 0 .7 into bracket 0 .07 plus 0 .08 divided by 2 so that is a 0 .07 plus 0 .08 divide by 2 that gives us 0 .075 and when we add that with 0 .7 we have this to be 0 .775 so put a negative sign so that means our z is minus 0 .775 excuse me so let's substitute that we have minus 0 .775 is equals to x minus our meal is 250 provided by 48 so minus 0 .75 times 48 that gives us minus 37 .2 is equal to x minus 250 so x is going to be cost to minus 7 .2 so x is going to be close to minus 7 .2.
05:15
2 plus 250 and that gives us 212 .8 so for our second question the answer is 212 .8 for our third question the question says uh oh excuse me i read the third question as the second question so i'm going to solve the second question right now as my third question and that is x is less than 19 percent of the data set x is less than 19 percent so i'm solving question too as my question 3 so 19 % so we have this to be 19 % over here 0 .19 % so let's go into our table the table towards the left so 19 % should be somewhere around here let's just try to set that so 19 % is going to be between this value and this value and let's trace that back we have a 0 .8s and if we trace it up we have 0 .70 it also so z is equals to 0 .08 plus 0 .07 plus 0 .08 are divided by 2 so z is the cost to 0 .8 plus so 0 .07 plus 0 .08 whereby so that gives us 0 .075 and do the math we have a 0 .875 so that means our z is minus 0 .0875.
06:56
So that means said we have that z is equal to x minus mule divided by zigma so that means a minus 0 .875 is equals to x minus 250 divided by 24 excuse me let's cross check that all right so that's that's divided by 48 rather this is 48 so minus 0 .875 times 48 that gives us a minus 42 is a to x minus 250 and when we do the math x is equal to minus 42 plus 250 and that gives us 208 so for the third question which is my question b that's 108 and uh so i call this question a this is question b and this is question c and for the first question we have that to be greater than 60 percent of the values so greater than 60 percent so we have something of this nature.
08:02
So 60 % is going to be somewhere around here.
08:06
Greater than 60%.
08:10
Excuse me.
08:11
We have a correspondence.
08:15
Let's try to have a better sketch.
08:21
So greater than 60%...