# 3.5 - Finding Cumulative Probabilities

3.5 - Finding Cumulative Probabilities#### Using the Standard Normal Table in the appendix of textbook or see a copy at Standard Normal Table

Table A.1 in the textbook gives normal curve cumulative probabilities for standardized scores.

- A
**standardized score**(also called*z*-score) is \(z=\frac{value-mean}{s.d.} = \frac{x-\mu}{\sigma}\). - Row labels of Table A.1 give possible
*z*-scores up to one decimal place. The column labels give the second decimal place of the*z*-score.

The cumulative probability for a value equals the cumulative probability for that value's *z*-score. Here, probability speed less than or equal 73 mph = probability z-score less than or equal 1.60. How did we arrive at this z-score?

**Example **

In our vehicle speed example, the standardized scores for 73 mph is

\(z=\frac{73-65}{5}=1.60\).

We look in the ".00" column of the "1.6" row (1.6 **plus** .00 equals 1.60) to find that the cumulative probability for z = 1.60 is 0.9452, the same value we got earlier as the cumulative probability for speed = 73 mph.

**Example**

For speed = 60 the z-score is

\(z=\frac{60-65}{5}=-1.00\).

Table A.1 gives this information:

The cumulative probability is .1587 for *z* = -1.00 and this is also the cumulative probability for a speed of 60 mph.

**Example**

Suppose pulse rates of adult females have a normal curve distribution with mean μ =75 and standard deviation s = 8. What is the probability that a randomly selected female has a pulse rate **greater than 85 **? *Be careful *! Notice we want a "greater than" and the interval we want is entirely above average, so we know the answer must be less than 0.5.

If we use Table A.1, the first step is to calculate a z-score of 85.

\(z=\frac{85-75}{8}=1.25\)

Information from Table A.1 is

Use the "05" column to find that the cumulative probability for z = 1.25 is 0.8944.

This is not yet the answer. This is the probability the pulse is less than or equal to 85. We want a greater than probability so the answer is:

P(greater than 85) = 1 - P(less than or equal 85) = 1 − 0.8944 = **0.1056. **

**Finding Percentiles **

We may wish to know the value of a variable that is a specified percentile of the values.

- We might ask what speed is the 99.99 th percentile of speeds at the highway location in our earlier example.
- We might want to know what pulse rate is the 25 th percentile of pulse rates.

To calculate percentiles in Minitab:

- Open Minitab without data.
- From the menu bar select Calc>Probability Distribution> Normal.
- Select the radio button for Inverse Cumulative Probability
- In the text box for Mean enter 65
- In the text box for Standard Deviation enter 5
- Since we do not have a column of data select the radio button for Input Constant and enter 0.9999
- Click OK
- The output is as follows:

To calculate percentiles in SPSS:

- Open SPSS without data.
- Because SPSS will not let you do anything without data just type something into the first blank cell (e.g. enter the number 2 in the first cell in column 1) and be sure to then click any other cell. You need to do this to complete the entry of the value into that cell.
- From the menu bar select Transform > Compute Variable
- In the box for Target Variable enter any name (e.g. percentile).
- Click inside the box for Numeric Expression (this should put the cursor inside this box)
- From the drop down menu for Function Group select Inverse DF
- From the list of Functions and Special Variables select IDF.Normal
- Click on the arrow next to the Delete button. This should put in the expression window the following: IDF.NORMAL(?,?,?)
- Replace ? with 0.9999,65,5 These values represent the cumulative probability value (0.9999 or 99.99 percentile), the mean (65) and standard deviation (5). BE SURE TO INCLUDE THE COMMAS AND KEEP THE PARENTHESES!!
- Click OK
- In the Data worksheet you should see a column with the target label (e.g. percentile) with the value 83.60 (If you click this cell you will see the in a text field above the cell the value of 83.59508242727854 but the number is rounded to two decimals in SPSS worksheet.)
- If you used the above labeling the worksheet would look as follows:

VAR00001 percentile 2.00 83.60

Note:

- The 99.99 th percentile of speeds (when mean = 65 and standard deviation = 5) is about 83.6 mph. Output from Minitab follows. Notice that now the specified cumulative probability is given first, and then the corresponding speed.

- The 25 th percentile of pulse rates (when μ = 75 and s= 8) is about 69.6. Relevant Minitab output is: