6.2 Using the Normal Distribution
Suppose that a random variable [latex]X[/latex] is normally distributed with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex]. The area under the normal curve for any interval of values of the random variable [latex]X[/latex] represents either the:
- proportion of the population with the characteristic described by the interval of values
OR - probability/chances that a randomly selected individual from the population will have the characteristic described by the interval of values.
Normal Distribution Probability Calculations
There are going to be two types of calculations/operations we will need to be familiar with when working with probabilities and normal distributions.
- Find AREA/PROBABILITY given VALUE
Given [latex]z-[/latex]score or [latex]x-[/latex]value, find the associated probability (area) - Find VALUE given AREA/PROBABILITY
Given a probability (an area), find [latex]z-[/latex]score or [latex]x-[/latex]value
Practice
EXAMPLE
The amount of time it takes a kidney stone to pass is distributed normally with a mean of 16 days and a standard deviation of 5 days. Suppose that one individual is randomly chosen. Let [latex]X =[/latex] time to pass the kidney stone. Round all answers to 4 decimal places where possible.
- What is the distribution of [latex]X[/latex] ?
- Find the probability that a randomly selected person with a kidney stone will take longer than 14 days to pass the stone.
- What is the cut off number of days for the bottom 10% of the people?
- Find the minimum number for the upper quarter of the time to pass a kidney stone.
SHOW SOLUTION
1. The random variable follows a normal distribution, we can write: [latex]X \sim N(16,5)[/latex]. For the rest of the questions, we’ll need to use technology.
DESMOS CALCULATOR
Calculator Usage Guide
In the first input box on Desmos calculator, enter: \[X=\text{normaldist}(16,5)\] and click on the magnifying glass icon to see the normal curve.
2. Greater than 14 days …
Since we want the probability/area, select the checkbox for Find Cumulative Probability (CDF) and enter 14 for Min. Leave the Max blank. Area is shaded and the answer is displayed. Area/Probability [latex]=0.65542174161[/latex].
Since we want the probability/area, select the checkbox for Find Cumulative Probability (CDF) and enter 14 for Min. Leave the Max blank. Area is shaded and the answer is displayed. Area/Probability [latex]=0.65542174161[/latex].
3. Bottom 10% cut off …
In input box #2, enter: \[X.\text{inversecdf}(0.10)\] Answer (displayed underneath the entry) = [latex]9.59224217228[/latex].
In input box #2, enter: \[X.\text{inversecdf}(0.10)\] Answer (displayed underneath the entry) = [latex]9.59224217228[/latex].
NOTE: The corresponding area/cutoff is not graphed when using X.inversecdf.
4. Minimum number for the Upper quarter…
Inversecdf takes area to the left. If the area to the right is 25%, then the area to the left must be 75%. In input box #3, enter: \[X.\text{inversecdf}(0.75)\] Answer (displayed underneath the entry) = [latex]19.372448751[/latex].
Inversecdf takes area to the left. If the area to the right is 25%, then the area to the left must be 75%. In input box #3, enter: \[X.\text{inversecdf}(0.75)\] Answer (displayed underneath the entry) = [latex]19.372448751[/latex].
Show me the steps for STATKEY
Click on NORMAL under Theoretical Distributions on the main Statkey page.
Click on Edit Parameters and enter your MEAN and STANDARD DEVIATION. For this question, mean [latex]=16[/latex] and standard deviation [latex]=5[/latex].
Click on Edit Parameters and enter your MEAN and STANDARD DEVIATION. For this question, mean [latex]=16[/latex] and standard deviation [latex]=5[/latex].
2. Probability that a randomly selected person with a kidney stone will take longer than 14 days to pass the stone [latex]= P(X > 14)[/latex]
Greater than 14 days. So we are given days which is a value of the random variable [latex]X[/latex]. We’ll need to find the probability, which is to say we need to find the area under the curve to the right of 14 on the horizontal axis. Select Right Tail at the top left. The blue textbox under the horizontal axis is for
[latex]x-[/latex]value and the the blue box in the middle is the area of the region shaded in red. Since we have [latex]x-[/latex]value of 14 days, click on the blue box below the number line and change that to 14 and hit enter. The area of the red region displayed representing more than 14 days should equal 0.655.
Greater than 14 days. So we are given days which is a value of the random variable [latex]X[/latex]. We’ll need to find the probability, which is to say we need to find the area under the curve to the right of 14 on the horizontal axis. Select Right Tail at the top left. The blue textbox under the horizontal axis is for
[latex]x-[/latex]value and the the blue box in the middle is the area of the region shaded in red. Since we have [latex]x-[/latex]value of 14 days, click on the blue box below the number line and change that to 14 and hit enter. The area of the red region displayed representing more than 14 days should equal 0.655.
3. Cut off number of days for the bottom 10% of the people…
Bottom [latex]10%[/latex] indicates the left tail area. Deselect Right Tail checkbox and then select Left Tail. Click on the blue box representing area (in the middle of the graph) and change that to [latex]0.1[/latex]. When you hit enter, the value below the number line will be your cut off point. Answer: 9.592
Bottom [latex]10%[/latex] indicates the left tail area. Deselect Right Tail checkbox and then select Left Tail. Click on the blue box representing area (in the middle of the graph) and change that to [latex]0.1[/latex]. When you hit enter, the value below the number line will be your cut off point. Answer: 9.592
4. Minimum number for the Upper quarter…
Upper quarter is the location where [latex]25%[/latex] of the values are to the right of the quartile value. So, we have the proportion of values to the right of [latex]25%[/latex] or [latex]0.25[/latex], and we need to find that cut off point. Deselect Left Tail checkbox and then select Right Tail. Click on the blue box representing area (in the middle of the graph) and change the value to 0.25 if it is not so already. When you hit enter, the value below the number line will be your cut off point. Answer: 19.372
Upper quarter is the location where [latex]25%[/latex] of the values are to the right of the quartile value. So, we have the proportion of values to the right of [latex]25%[/latex] or [latex]0.25[/latex], and we need to find that cut off point. Deselect Left Tail checkbox and then select Right Tail. Click on the blue box representing area (in the middle of the graph) and change the value to 0.25 if it is not so already. When you hit enter, the value below the number line will be your cut off point. Answer: 19.372
When you are familiar with the above example, check your answers using SUBEDI Calculators as well. This particular version of the calculator is actually Desmos in the background doing all the heavy-lifting.
