Normal distribution

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Deannnn97
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#1
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A Normal distribution has a mean of 50, and the proportion of the distribution less than 40 is 2.5 %.
(a) What is the standard deviation of this distribution? [3 marks]
(b) What is the probability of a value greater than 60? [2 marks]

PLEASE HELP
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gdunne42
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(Original post by Deannnn97)
A Normal distribution has a mean of 50, and the proportion of the distribution less than 40 is 2.5 %.
(a) What is the standard deviation of this distribution? [3 marks]
(b) What is the probability of a value greater than 60? [2 marks]

PLEASE HELP
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A. There are ways to calculate this exactly but I assume you are being asked this question to test understanding of the basic normal distribution curve. Approximately 95% of data lies within 2 standard deviations above and below the mean. B. the curve is symmetrical about the mean.


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RDKGames
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(Original post by Deannnn97)
A Normal distribution has a mean of 50, and the proportion of the distribution less than 40 is 2.5 %.
(a) What is the standard deviation of this distribution? [3 marks]
(b) What is the probability of a value greater than 60? [2 marks]

PLEASE HELP
In the case you need to work these out:

(a) Lay out the information first;

X ~ N(50,\sigma^2)

Therefore:

z=\frac{X-50}{\sigma}

Next you are told that the probability that the proportion of the distribution less than 40 is 2.5% thus:

P(X<40)=0.025

which looks like:

Image

and due to symmetry around the mean;

Image

Which means that:
P(X<40)=P(X>60)=0.025

P(X>60)=1-P(X<60)

P(X<60)=0.975

From here we need to convert out probability into the z value by means of the normal distribution table;
\phi(0.975)=1.96=z

and now we can sub it into our formula for z and solve for \sigma; which is the standard deviation.

(b) We found it in part (a) as part of out working out
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Deannnn97
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thanks man helps so much, just the grey areas in the diagram, how do you get this from the 2.5%?
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RDKGames
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(Original post by Deannnn97)
thanks man helps so much, just the grey areas in the diagram, how do you get this from the 2.5%?
Do you mean the corresponding z value from 2.5%? Just refer to the normal distribution data table for that.
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Deannnn97
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(Original post by RDKGames)
Do you mean the corresponding z value from 2.5%? Just refer to the normal distribution data table for that.
No i mean the grey area, from 40 down and 60 up, how does this represent 2.5%? I get 40 as this is part of the question, and so you copy this onto the otherside making it 60, but where how does the 2.5% effect it, for example if the question was; the proprtion of distribution less than 40 is 1%, how would this differ?
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RDKGames
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(Original post by Deannnn97)
No i mean the grey area, from 40 down and 60 up, how does this represent 2.5%? I get 40 as this is part of the question, and so you copy this onto the otherside making it 60, but where how does the 2.5% effect it, for example if the question was; the proprtion of distribution less than 40 is 1%, how would this differ?
Well the question outlines this percentage for us, saying less than 40 is 2.5% so that's what I represented on the graph; everything less than 40 is 2.5%. If you were to change only the percentage, you would be altering the standard deviation.
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Deannnn97
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(Original post by RDKGames)
Well the question outlines this percentage for us, saying less than 40 is 2.5% so that's what I represented on the graph; everything less than 40 is 2.5%. If you were to change only the percentage, you would be altering the standard deviation.
ok so a) would be 1.96
b) would be 0.975%?
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RDKGames
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(Original post by Deannnn97)
ok so a) would be 1.96
b) would be 0.975%?
No and no.The z value is not the standard deviation. Probability it's more than 60 is the same as probability its less than 40.
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Deannnn97
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(Original post by RDKGames)
No and no.The z value is not the standard deviation. Probability it's more than 60 is the same as probability its less than 40.
So what are the answers?
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RDKGames
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(Original post by Deannnn97)
So what are the answers?
Well I'm not here to solve your questions for you. Think. I literally did most of the working out for you in (a) and answered b as part of working out in (a).
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