Normal distribution

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

Reply 1

gdunne42

21

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

ImageUploadedByStudent Room1470139823.447050.jpg

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.

Posted from TSR Mobile

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:

and due to symmetry around the mean;

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

(edited 7 years ago)

Reply 3

Deannnn97

OP

thanks man helps so much, just the grey areas in the diagram, how do you get this from the 2.5%?

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.

(edited 7 years ago)

Reply 5

Deannnn97

OP

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?

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.

Reply 7

Deannnn97

OP

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%?

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.

Reply 9

Deannnn97

OP

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?

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).

Normal distribution

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