Statistics Stanadard Deviation Help Urgently please !!

x

I would suggest getting the textbook, reading through them and attempting the questions and post here with your working if you're stuck

I don't have a textbook

Get a hold of one if you can, they are really useful.

See the link in the above post for some videos, or find some if that's not a useful one.

Get a hold of one if you can, they are really useful.

See the link in the above post for some videos, or find some if that's not a useful one.

so for the first one I did

= P(10-16/3 < Z < 12-16/3)
= P(-2 < Z < -1.33)

now do I do

= 1- P(Z > 1.33) - 1 - P(Z > 2)
??
I'm stuck

Good start

So you have P(-2 < Z < -1.33), which can be written as P(Z<-1.33) - P(Z<-2). Why? You may wish to use a number line to help you.

You can split the range Z < -1.33 into two different regions: -2 < Z < -1.33 (which is what we want) and the rest is Z<-2, and combining those two regions together again gives Z < - 1.33. So if we express P(-2 < Z < -1.33) in terms of P(Z < - 1.33) then we would be counting the area 'Z < -2' but we're not interested in that, only the region -2 < Z < -1.33, so that's why it can be written as P(Z<-1.33) - P(Z<-2). That's just for your information/understanding.

Now the problem is that the tables have values P(Z<z) where z is a positive number, and you have to use properties of the normal distribution when you're dealing with things like P(Z>z) or P(Z<-z) (where z is a positive number) or even P(Z>-z).

So we've got P(Z<-1.33) - P(Z<-2) to work out. We need everything to be of the form P(Z<z), where z is a positive number.

So you use two things (one of which you've done already). P(Z<-z) = P(Z>z) by symmetry. This is because if you draw the graph it's symmetric around 0.

The other rule that you can use is that P(Z>z) = 1-P(Z<z), which you can kind of mean as (if it's not less than z then it's more than z or vice versa). Just like 'if the probability that it's going to rain is 0.2, then the probability that it won't rain is 0.8 (1-0.2)'.

So you've tried to use both of those things but you've got to keep track of how you do them.

Good start

So you have P(-2 < Z < -1.33), which can be written as P(Z<-1.33) - P(Z<-2). Why? You may wish to use a number line to help you.

You can split the range Z < -1.33 into two different regions: -2 < Z < -1.33 (which is what we want) and the rest is Z<-2, and combining those two regions together again gives Z < - 1.33. So if we express P(-2 < Z < -1.33) in terms of P(Z < - 1.33) then we would be counting the area 'Z < -2' but we're not interested in that, only the region -2 < Z < -1.33, so that's why it can be written as P(Z<-1.33) - P(Z<-2). That's just for your information/understanding.

Now the problem is that the tables have values P(Z<z) where z is a positive number, and you have to use properties of the normal distribution when you're dealing with things like P(Z>z) or P(Z<-z) (where z is a positive number) or even P(Z>-z).

So we've got P(Z<-1.33) - P(Z<-2) to work out. We need everything to be of the form P(Z<z), where z is a positive number.

So you use two things (one of which you've done already). P(Z<-z) = P(Z>z) by symmetry. This is because if you draw the graph it's symmetric around 0.

The other rule that you can use is that P(Z>z) = 1-P(Z<z), which you can kind of mean as (if it's not less than z then it's more than z or vice versa). Just like 'if the probability that it's going to rain is 0.2, then the probability that it won't rain is 0.8 (1-0.2)'.

So you've tried to use both of those things but you've got to keep track of how you do them.

Consider the distribution X~N(15,16) to calculate the following probability:
The value of a if P(X<a) = 0.975

would I start by doing : P(Z < a - 15 / 16 ) = 0.975?

of do I divide by 4 ??

Swap the 16 for 4 and you are correct remember that the second number is variance, and the square root of that is the standard deviation, so you could write it as N(15,(4^2)) just to remember that the standard deviation is 4.

What next?

the do u just do

a - 15 / 4 = 0.975
so
a - 15 = 3.9
add 15
a = 18.9 ??

sorry ive never done a question like this before

Remember when we're working in the opposite direction, we start with an x value, turn it into a z value (so go from P(X<x) to P(Z<z), and find the probability that matches up with that z value.

This time (in the opposite direction) we start with the probability, so what would we need to find out?

P( Z > a ) = 0.975
Z = ??

I have no idea what I'm doing

So we've got that P(Z < (a - 15) / 4 ) = 0.975

Usually when we know what a is and we want to find out what the probability is (in this case 0.975 would be our final answer), so we take a, turn it into a z value using z = (a-mean)/standard deviation, then since we have P(Z<z) we go to the tables and find our answer, which would be 0.975 in this example.

But like I said last time we are working backwards... so we have the probability and we need to find out what a is. It is 'reversing' the steps, so what do you think we should do?

would we find 0.975 in the table to find z which is 1.96

Correct so what does that z equal to?

1.96.

So how to I work out a?

How will z=1.96 be related to a?

Think about how you calculate a given z and work backwards.

so
P(1.96 < a) = 0.975 ?!