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# S2 Normal dist - IQ watch

1. Intro to question states distribution of IQ amongst children of a certain age group can be approximated by Normal distribution and mean=100, sd=15. You are given that phi(2.5) = 0.994.

A class of 30 children is selected at random from the age group. Calculate to 3 s.f. the probability that at least one member of the class has an IQ of 138 or more.

Here is my working.

IQ measures are discrete so we use continuity corrections to correct for this when using Normal distribution. So 138 or more becomes 137.5 or more (to include 138 +- .5

Therefore, z calc is:

z = (137.5 - 100) / 15 = 2.5

2.5 is value mentioned in question above so I assumed just use their value of 0.994.

Hence, probability of an IQ of 138 or above = 1- 0.994 = 0.006

Now there are 30 people in class. So I just multiplied by 30.

30 * 0.006 = 0.18

However, answer in book is 0.165.

What have I done wrong?
2. (Original post by acomber)
Intro to question states distribution of IQ amongst children of a certain age group can be approximated by Normal distribution and mean=100, sd=15. You are given that phi(2.5) = 0.994.

A class of 30 children is selected at random from the age group. Calculate to 3 s.f. the probability that at least one member of the class has an IQ of 138 or more.

Here is my working.

IQ measures are discrete so we use continuity corrections to correct for this when using Normal distribution. So 138 or more becomes 137.5 or more (to include 138 +- .5

Therefore, z calc is:

z = (137.5 - 100) / 15 = 2.5

2.5 is value mentioned in question above so I assumed just use their value of 0.994.

Hence, probability of an IQ of 138 or above = 1- 0.994 = 0.006

Now there are 30 people in class. So I just multiplied by 30.

30 * 0.006 = 0.18

However, answer in book is 0.165.

What have I done wrong?
If there is at least one with an IQ of 138 or above then they are not all below 138.

You can calculate 1-P(all below 138).
3. (Original post by BuryMathsTutor)
If there is at least one with an IQ of 138 or above then they are not all below 138.

You can calculate 1-P(all below 138).
OK, so the probability that all 30 students are below 138 is: 0.994^30 ?

1 minus P(all below 138):

Is that correct?
4. (Original post by acomber)
OK, so the probability that all 30 students are below 138 is: 0.994^30 ?

1 minus P(all below 138):

Is that correct?
You got it. Are you happy with that now?

It's often helpful to think of "at least one" as "not none".
5. (Original post by acomber)
OK, so the probability that all 30 students are below 138 is: 0.994^30 ?

1 minus P(all below 138):

Is that correct?

Great Success !!!
6. (Original post by BuryMathsTutor)
You got it. Are you happy with that now?

It's often helpful to think of "at least one" as "not none".
Yes, thank you. I will burn your tip about at least one == not none into my brain!

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