find the 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.

Answer 1

Hello again, Madeline,

As with our other problems we worked together, this is the exact same type of problem. As a tutor, I feel a responsibility to tell you exactly what we are doing when calculating these scores, otherwise you will be able to calculate a score but not understand what that calculation means.  
  
So, if you will indulge me here for a moment: A z score is a transformation of a given raw value which precisely tells you where that given raw value is in relationship to the mean, given the distribution of values are normal (as they have been with all of the problems). In addition, that z score allows us to know the probability of a given value occurring given the conditions outlined in the problem. You will notice that the probabilities always fall along the x-axis of the Normal Bell Curve. Also, if the distribution is normal that means your bell curve is also normal and has the following characteristics: 

(1)  It is symmetrical 
(2)  It has two asymptotic tails which means the tails continuously approach the x-axis without ever touching
(3)  The mean, median and mode are ALL the same. 

Again: All we are doing with your problems is transforming the raw values to scores allowing us to determine the probability of the outcome in question (whether that be pulling a specific rod out of a batch, or pulling a single thermometer out of a batch). Are we good? Of course we are, because you are superhero of z-scores now. 

So, let's solve the problem. 

Remember a z-score formula is as follows: 

z = x - xbar
     -----------
        sd 

Whereby, x is the raw value, xbar is the mean and sd is the standard deviation. With your information given in the problem the following applies: 

z = -2.9855 - 0
      ------------

         1.00

z = -2.9855
      ---------
         1.00

z = -2.9855   


Statisticians, including those of us in psychology all have divergent opinions regarding what to do with the above z score but I go by the policy of "four and below; let it go. Five and above; give a shove," meaning I would round that z-score to the following: 

z = -2.99 (For some reason I believe your instructor will agree). 

Remember our chart of z-score probabilities? Here is the link: http://www.z-table.com/ 

Please remember we are going to use the first chart because it provides probabilities for negative z scores. 

You will follow the exact same process as the other two problems. First, use the first column and go down to -2.9 and when you locate that row, go across to the column which is .09. The intersecting value is your four decimal answer which is:

.0014  

The probability of that thermometer being randomly selected and tested while obtaining a reading less than -2.985 degrees celcius, is therefore: 

.14% Remember all you do is move your decimal over two places to the right. 


Again, please note that this value encompasses ALL values to the left and so you do not have to do any further calculations. 

I realize most students just want the answer, and in statistics this can be a very deadly strategy because if you do not know the why behind the how, the subject matter becomes extremely confusing and I as a tutor, would be setting you up for failure later down the road should you end up taking futher statistics classes. Now you know the how and why of this particular z-score. 

I hope you have an excellent weekend.