Question

to find a probability of values in a tail of the normal distribution using a standard normal table

Answer 1

Solution: How to find a probability of values in a tail of the normal distribution using a standard normal table?

Answer:

It will be better to illustrate this using an example:

Let's suppose a process follows a normal distribution with mean and standard deviation

Also let's suppose, we have to find the

To find this probability, we first need to convert into standard z score using the below formula:

We have got z-score = -1

Now we can express , which means 45 is 1 standard deviation less than mean 50.

Now to find the , we will use standard normal table.

In the above left tailed table (with negative values), grey shaded first column and grey shaded top row together make z score values. And rest values in table denote the probabilities to each corresponding z score. For example -1.5 in first column and 0.06 in the first row will make -1.56 and its corresponding probability will be 0.0594.

Now using the same logic, we have:

Therefore the probability of x less than 45 is 0.1587.

Now let's consider another situation, where we have to find the

Here z score will be:

  

So here we have to find the

We know that using the complementary law:

Now using the left tailed table (with positive values), we have:

We clearly see that:

Therefore of x greater than 55 is 0.1587.

Please note in both the cases, we got Probability 0.1587. This is because the normal distribution is symmetric about mean. That is, the area to the left of the mean is same as the area to the right of mean.

Hope this helps!