Question

18. What proportion of values in a normal distribution will be above 2.33 standard deviations to the right of (or above) the mean?

Answer 1

Normal distribution curve with mean '' and variance '² ' is represented as:

The standard normal distribution is:  

P(Z < z) is the probability of values of Z falling less than z. Where 'z' is a constant.

P(-z < Z < z) is the probability of values of Z falling between -z and +z.

Here ' z ' on a standard normal distribution curve implies z standard deviations from the mean on a normal distribution curve.

So for example the probability of Z less than 1.5, P( Z < 1.5 ) represents the proportion of values falling less than 1.5.

To find out the exact proportion of values falling above 2.33 deviations to the right of the mean we need to find out P(Z > 2.33)

To find out P(Z > 2.33) we need to first find out P( Z < 2.33 ) and subtract this from the total probability, i.e 1.

Standard Z - tables gives the area under the curve starting from the left most point of the table till the required point.

From the standard Z - tables,

P(Z < 2.33) = 0.9901

Total probability = 1

1 - P(Z < 2.33) = P(Z > 2.33)

=> P(Z > 2.33) = 1 - 0.9901

=> P(Z > 2.33) = 0.0099

Therefore, proportion of values falling above 2.33 standard deviations to the right of the mean is 0.99% .