Solving Problems that Involve the Normal Distribution - Graphics require Java
Do you have JAVA Plug-in Installed?
The following will calculate probabilites but has no graphics
Calculation of Probabilities Using the Normal Distribution
If the problem involves one z value
Input z values can vary from -3 ro +3 and may have up to three digits after the decimal point. For example any of the following would be valid entries: 2., 2.5, 2.57 and 2.575 The model assumes that virtually all of the area under the curve lies between -3 and +3 standard deviations from the mean. For most applications this is a reasonable safe assumption. In checking this tool against the tables in statistics textbooks I found that this tool usually gives answers that are with a few ten-thousandths of the values given in the books. In general the further you are from the mean the better the agreement.
- Calculates P(z < some value)
- and P(z > some value)
- Except for rounding errors values should add to 1.00
- If the problem involves two z values
- Calculate P(z < given value one)
- P(given value 1 < z < given value 2)
- P(z > given value 2)
- Except for rounding errors values should add to 1.00
Working with the normal probability distribution should be easy! It is a very well behaved distribution and the calculations are relatively simple. The hardest part usually involves reading the problem and reading the tables in the textbook. If all goes well this tool will help you understand what is going on and allow you to read and understand the tables like a pro. To use this tool you must have Java Installed. If you don't have Java and don't want it there is a companion tool that doesn't use graphics that you can use. Frankly I don't think it is as good as this one (and I wrote it) but it may do the trick for you.
The usual question in statistics involves determining the probability of something. For a continuous distribtion such as the Normal the area under the curve represents probability. Problems usually ask you to determine the probability that some value will be between two other values. Examples of questions are:
- Determine P(z > 1.61), Determine the probability that a z value is more than 1.61
- Determine P(X < 143), Determine the probability that the variable X is less than 143
- Determine P(-1.2 < z < 2.3), Determine the probability that z is between -1.2 and 2.3
- Determine P(z < -1.5 or z > 0.75), Determine the probability that z is less than - 1.5 or more than 0.75
The figure on the right provides you with a graphical view of some questions that you could be asked. The table in your textbook starts from the left-tail and goes to the right-tail. As it goes it gives you the cumulative probability to the current point. The point is defined by a z value. Virtually all of the area under the curve will be between z = -3 and z = +3. In otherwords from 3 standard deviations below the mean to 3 standard deviations above the mean. The tables can sometimes be confusing. A picture usually helps. Here is what the tool will do for you:
- It will graph the normal probability distribution for you
- It will put a vertical line at one or two critical values of z based on your inputs
- It will write out the probability below a critical z value,
- the probability between z values (if there are two) and
- the probability above the uppermost z value
To do the things listed above you will have to provide the tool with the following information. Use the input field to provide this information:
- If it is a z value problem does the problem involve one or two z values?
- For one value enter the single z value and hit the enter key.
- For two z values enter the first z value a comma, the second z value and then hit the enter key.
- If you have population data instead of z values. For population values calculate the z value using z = (x - μ)/σ
- If you have sample data instead of z values. Calculate the z value using z = (x - μ)/(σ/√ n)
- Now enter the z values in the tool just as you would have if you had been given z values all along
- The screen should display a normal distribution, one or two vertical bars at critical values of z and
- Probability values in each of the regions of the graph.
Let's use the tool to solve the problems shown on the right.
- Enter 0.82 and hit the enter key
- The normal distribution is displayed with a vertical line marking the critical value of z
- To the left of the line you have Pr = 0.7939 meaning there is a probability of 0.7939 that you will get a z value less than 0.82
- To the right of the line you have Pr = 0.2061 meaning there is a probability of 0.2061 that you will get a z value more than 0.82
- Only the 0.7939 value is in the tables in your textbook. To get 0.2061 you subtract 0.7939 from 1.0
- The total area under the curve is 1.0 so the area to the right of the critical line is 1 - 0.7939 = 0.2061
The tool also has a print capability. If you click on the print button at the bottom of the graph it will set up a print file. If you are doing this over the web and have security flags set on your browser you will probably be asked if it is ok for the Applet to print something. If you want the graph you should answer yes.
What if it is not a z value problem? Here are the problem codes that the tool can handle
- Z1, Z2 - one or two critical z values. For this problem enter theZ1 or Z2 code then a comma a z value a comma and then if required a second z value
- One z value - 2.3 or Two z values - 1.4, 2.1