Question

IQ scores in a large population have a normal distribution with mean=100 and standard deviation=15. What is the probability the sample mean for n=2 will be 121 or higher?

(I understand the z score equals 1.98,could you please explain to me how the final answer is equal to 0.0239?thanks.)
Also, why do we use 1 and subtract 0.9761 to get 0.0239? Where does 0.9761 come from?

Answer 1

It appears that you have already done the calculation for the z value. So I will skip that part. I will tell you the scenarios on which you should use which table and how to calculate the probability from z value.

In this case the distribution increases from left to right. This means consider a normal distribution chart like the one shown below. Left hand side of the curve will begin from 0 and will eventually move to the highest level. Now, keep in mind, that the normal distribution is usually an infinite chart. However, we usually represent it within 3 standard deviations. Anyway, the basic concept is that it increases as it goes from left to right. This is also known as a right tailed distribution table.

There are two types of tables, right or left tailed. The most commonly used are the right tailed ones and we can use either of them. For this example, we will the right tailed chart (increases in value from left to right).

Now the question has asked for people who have IQ 121 or more. This means in the distribution chart these numbers will fall on the right most corner beyond the point where I have marked 121. These marking is only symbolical and not accurate.

In a right tailed chart, this means that in order to find people who are IQ 121 and higher, we need to discard people who has IQ lower than 121. In a right tailed chart the population is always represented on the left hand side of the value. This means, when we calculate the z value and go to the normal distribution table, that table tells us the population on the left hand side of z value.

In other words the value that we get by using a right tailed table will be shaded from 0 to 121 (in this case). So ideally we end up finding out the people whose IQ is below 121.

With a z value of 1.98 we find the probability to be 0.9761. This you can obtain from a z table (or a normal distribution table). Now since we are using a right tailed chart, instead of telling us the probabilities for 121 and above, it is telling us below 121. This means that the probability of people below an IQ of 121 (it has a z value of 1.98) is 0.9761.

I hope this is clear till this point.

The basic rule of probability is that if we combine all the probabilities, the value is always 1. It cannot be more or less. So if probability of people below IQ of 121 is 0.9761 then the probability of people with IQ of 121 and above is the remaining value from 1. This is why we subtract 1-0.9761 in order to arrive at 0.0239.

For exercise look at the table below. Add the column header to the row header in order to get the corresponding probability value. I have marked the point. Hope this is clear. You will get this table on your text book or online. They are standard tables. However, just be careful about left or right tailed. Otherwise you will get an opposite answer. In this case these are right tailed tables.