Finding the percentage from the normally distributed mean and standard deviation? Help please :)

Question

So, this is the question I'm have difficultly working out, I've come close to the answer but my calculations are clearly going wrong, anyway, here's the question and the help would be much appreciated. 

 

Sevenday, a retailing group, has analysed the monthly spend by its loyalty card customers and found that it is normally distributed with a mean of £130 and a standard deviation of £48. The percentage of customers that spend more than £100 per month is?

 

 

 

Answer 1

Jamie:

 

You wish to find out "The percentage of customers that spend more than £100 per month."

 

Let X denote the amount a customer spends.  You are given that X is normally distributed with mean BP 130 abd SD BP 48,  You want the probability P[ X > BP 100].

 

Step 1: Calculate the z_value. The formula is z = (individual value - mean) / standard_dev.  So z = (100 - 130)/48

= - 0.625.  You want the area under the original normal curve with mean 130 and std_dev 48 to the right of 100.

 

That area is the same as the area to the right of z = - 0.625 under the standard normal curve (mean 0, std_dev 1).  The conversion to a z+value is necessary because the area under the standard normal curve have been tabulated.

 

Now, there are three ways of proceeding further. 

 

One is to refer to the appropriate table in your text book and get the area.  You may have to interpolate.

 

The second way is to use technology.  These areas are obtainable on advanced calculator, such as TI-84.  Refer to the manual to find out how to get the areas under the normal curve.

 

The third way is to use internet.  (I am going to show you this one because this will add to your knowledge in a unique way).  Go to the website  (this is freeware)

 

http://www.danielsoper.com/statcalc3/calc.aspx?id=2

 

Input z_score = - 0.625. Click on Calculate.  You will get 0.266 (to 3 decimals).  BE CAREFUL. THIS IS THE AREA TO THE LEFT OF the z_score.  You want the area to the right.  Since the curve is continuous, the required area is

 

1 - 0.266 = 0.734.

 

In context, the percentage of customers that spend more than £100 per month is 73.4,

 

Dattaprabhakar (Dr. G.)