finding probability

Question

Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading less than -2.985°C. Round to 4 decimal places.

P(Z<−2.985)=

Answer 1

Hello Paula,

My apologies for you not getting an answer earlier. I just was surfing through some of the content areas and saw your question, and will provide you an answer now, despite it probably being late. 


What you are seeking to determine is a z-score transformation. 

The formula for a z-score is as follows: 

z = x - xbar

     ---------

        sd 

where x = the raw value (in your case the reading of -.2.985), x bar is the mean, and sd is the standard deviation. 

I am assuming the value you are seeking is negative, but in the event I am incorrect, I will give you both for reference. 

z  = -2.985 - 0
      ------------

         1.00 

z =  -2.985
       --------

         1.00

 

z = -2.985 

To determine the probability you must use a z score table with both negative and positive values for easier reference. 

Here is a link to one such table: 

http://www.z-table.com/

Since your z score value is -2.985 most statisticians would round this up to -2.99 (four and below let it go; five and above give a shove). 

Go to your table and notice the first column is a z score column. Go down to -2.9 and follow that row across to the column which is .09. Please note: All you are doing is adding .09 to the -2.9 and this is the way you use the z score table. The intersecting value of the row -2.9 and the column .09 is the four decimal answer you seek: 

That value is .0014  To determine the probability you would multiply by 100 which moves your decimal to places to the right. That means the probability of you randomly selecting a thermometer with a reading less than -2.985 is .14%



If the value is positive: that is, if you are seeking what the probability of drawing a thermometer and obtaining a reading less than 2.985 you use the same process but use the second table which has positive z scores. I would still round the value to 2.99 and you look up the associated value in the same way you just did with the negative value. 

Locate the 2.9 row under the z column and follow across to the .09 column. The intersecting value is your four decimal answer which would be .9986 

To convert to a probability percentage, again just move your decimal two places to the right. The answer then becomes 99.86% probability that you would randomly select a thermometer with a reading less than 2.985. 

Hope this helps you and that you have a great weekend.