High school Normal distribution two questions

[synkk]

TITLE: Normal distributions (sorry)

Question 1:

Working:

Now the answer for sigma i got correct, but μ i got incorrect for some reason. Could anyone explain where I've gone wrong for μ?

Question 2:

Workings:

All i need help is with part c and d:

for c) i done 1 - 0.6463 to get the probability of the two shaded bits, but where do i go to get the part they are asking for?

for d) i have no idea, any help would be great.
thanks.

 

[HallsofIvy]

Well, one obvious error is that you have both $(25-\mu)/\sigma$ and $(64-\mu)/\sigma$ equal to -0.67. One of them should be positive. In any case, the normal distribution is symmertric so it should be clear that the mean will be exactly between the two "quartiles".

The two shaded bits? I assume you mean the one shaded bit and the other outlieing area. That, of course, is 1- .6463. To find the two parts separately, use the fact, as you are given, that upper piece has twice the area of the lower piece.

 

[synkk]

Well, one obvious error is that you have both $(25-\mu)/\sigma$ and $(64-\mu)/\sigma$ equal to -0.67. One of them should be positive. In any case, the normal distribution is symmertric so it should be clear that the mean will be exactly between the two "quartiles".

The two shaded bits? I assume you mean the one shaded bit and the other outlieing area. That, of course, is 1- .6463. To find the two parts separately, use the fact, as you are given, that upper piece has twice the area of the lower piece.

for question 1) Why is one of the positive (the second one) if they are symmetrical. Also how can i use use symmetry to find the mean? Is it 25+45 / 2?

for question 2) I still don't get how to get part C.

 

[Ray Vickson]

for question 1) Why is one of the positive (the second one) if they are symmetrical. Also how can i use use symmetry to find the mean? Is it 25+45 / 2?

for question 2) I still don't get how to get part C.

Is it 25+45 / 2? Close, but not quite: it IS (25 + 45)/2---what you wrote is 25 + 22.5. Brackets matter!

RGV

 

[synkk]

Is it 25+45 / 2? Close, but not quite: it IS (25 + 45)/2---what you wrote is 25 + 22.5. Brackets matter!

RGV

Could you explain why it is (25 + 45) / 2? I don't understand how the symmetry makes this true.

 

[Ray Vickson]

Could you explain why it is (25 + 45) / 2? I don't understand how the symmetry makes this true.

The distance between the mean and the 25th percentile is the same as the distance between the mean and the 75th percentile, because of symmetry. Therefore, the mean is the average of the 25th and 75th percentiles.

RGV

 

[synkk]

The distance between the mean and the 25th percentile is the same as the distance between the mean and the 75th percentile, because of symmetry. Therefore, the mean is the average of the 25th and 75th percentiles.

RGV

Alright thank you. Do you have any idea on how to do question 2 part c and d?

 

[Ray Vickson]

Alright thank you. Do you have any idea on how to do question 2 part c and d?

Yes, but I think you have already been given enough hints by others. Just look at the diagram, sit down, relax, and _think_. Don't try to write down the answer right away: approach it systematically, piece-by-piece, and don't worry if it takes longer than you think it should. After you see what is happening you can go back and clean it up.

RGV

 

[synkk]

Yes, but I think you have already been given enough hints by others. Just look at the diagram, sit down, relax, and _think_. Don't try to write down the answer right away: approach it systematically, piece-by-piece, and don't worry if it takes longer than you think it should. After you see what is happening you can go back and clean it up.

RGV

still on question 1) i don't understand why (45 - mu) / sigma is positive 0.67, could you explain please?

for c i done (1-0.6463)/3 as there is 3 parts, with the shaded region being the smallest, which gets me the correct value.

For D I've standardized it but I'm not sure what value i should put it against...

 

[Ray Vickson]

still on question 1) i don't understand why (45 - mu) / sigma is positive 0.67, could you explain please?

for c i done (1-0.6463)/3 as there is 3 parts, with the shaded region being the smallest, which gets me the correct value.

For D I've standardized it but I'm not sure what value i should put it against...

Never mind trying to understand whether (45 - mu) / sigma is 0.67, just tell me: what do YOU think the value of (45 - mu)/sigma should be? Remember, read the whole question carefully before answering.

RGV

 

[synkk]

Never mind trying to understand whether (45 - mu) / sigma is 0.67, just tell me: what do YOU think the value of (45 - mu)/sigma should be? Remember, read the whole question carefully before answering.

RGV

I think it should equal -0.67 because P(z>(45-mu)/sigma) = 0.75 meaning it is in the left hand side of the distribution, so we reflect it and we are trying to find P(z<(45-mu)/sigma) which is 0.67, but because we reflected it doesn't it mean it is negative i.e that originally z was in the left hand side so negative.

thanks for continuing to help.

 

[Ray Vickson]

I think it should equal -0.67 because P(z>(45-mu)/sigma) = 0.75 meaning it is in the left hand side of the distribution, so we reflect it and we are trying to find P(z<(45-mu)/sigma) which is 0.67, but because we reflected it doesn't it mean it is negative i.e that originally z was in the left hand side so negative.

thanks for continuing to help.

You should avoid just blindly using formulas you do not really understand. Instead: think! If 45 is the 75th percentile, and μ is the 50th percentile (also equal to the mean), we MUST HAVE 45 > μ, so (45 - μ)/σ must be > 0. It cannot possibly be -0.67, which is a negative number.
Your "equation" P(z>(45-mu)/sigma) = 0.75 is wrong, but would be OK if you replaced the 0.75 by ___ ? (I'm leaving it to you to fill in the blank.) Always, always, draw a picture.

RGV

 

[synkk]

by 0.25? so P(Z>(45-mu)/sigma) = 0.25 ? Is this correct?

 

[Ray Vickson]

by 0.25? so P(Z>(45-mu)/sigma) = 0.25 ? Is this correct?

Yes.

RGV

 

[synkk]

Yes.

RGV

Thank you for all the help, i done question 2 part d also.