00:01
So in the given question we are asked, we are told that we have a normal distribution and we have to answer the following questions.
00:10
So let's start with the first question.
00:13
So what i have over here is the figure of the standard normal distribution table.
00:19
The standard normal distribution table and the values that you would get from this table are actually the areas under the curve under the normal curve and the z value is 0 in the middle of the curve and if we choose for example a value that is like 1 .11 then this value is actually the corresponding area of let's assume this is the point then this is the point 1 .11 so the area that is the number that is given over here would be actually this area right so this is the value this is the kind of value that you get from the standard normal distribution table so what we can do is now answer each of the questions about the normal distribution that we have been asked in the question so starting from the first part first what we have been asked is what the score forms the boundary between highest 15 percentage and rest of the scores that score z score that forms that forms boundary boundary between between highest 15 percentage highest 13 percentage and the rest of the scores rest of the scores so what we would have over here is we can take a normal variable a normal varied let's assume that it is x or let's say that it is z itself right the normal variable is z and it should be greater than a value which is let's take capital z and small z so and the probability that it is greater than a certain value z is actually given as 15 percentage right so we can write this we can find this probability by taking 1 minus the probability of the value of the variable z to be less than small z and this would be equal to 0 .15.
03:26
Now we can write the probability that p of z is less than or equal to the small z is actually 1 minus 0 .15 which is 0 .85.
03:43
Now we take the normal distribution curve and what we can do is we have to find a value which is corresponding to 0 .85, right? so we are finding such a value which is in this part of the curve and this would be after before this point, the probability is 85 percentage and after this point it is 15 percentage right so what we can find is since the curve is symmetrical the half of the curve would have a probability of 0 .5 and the rest would have the probability of whatever is required over here so we can split this 0 .85 and write it as 0 .5 plus 0 .35 right 0 .5 is until this middle point and 0 .35 is over here.
04:50
So we just have to find the value that is corresponding to 0 .35 from the normal distribution table.
05:00
So 0 .5 plus 0 .35 is what you would have.
05:05
And when we check the table, we can see that there is the value 0 .3508.
05:13
Which we are going to take over here.
05:15
So this is actually 1 .14, right? so the required z value over here is that pf capital z plus that 1 .04 is actually equal to 0 .85.
05:36
So we found the required z value to the 1 .04 in the first case.
05:43
Now similarly, we can do the other ones too.
05:48
So in the other case, that is in the second part of the problem we are asked, the z score that forms the boundary between highest 40 percentage and the rest of the numbers, highest 40 percentage and the rest of the numbers.
06:10
So in a similar way we can write p of z, p of z...