3.3.2 - The Standard Normal Distribution | STAT 500 (2024)

A typical four-decimal-place number in the body of the Standard Normal Cumulative Probability Table gives the area under the standard normal curve that lies to the left of a specified z-value. The probability to the left of z = 0.87 is 0.8078 and it can be found by reading the table:

1. Since z = 0.87 is positive, use the table for POSITIVE z-values.
2. Go down the left-hand column, label z to "0.8."
3. Then, go across that row until under the "0.07" in the top row.

You should find the value, 0.8078. Therefore,\(P(Z< 0.87)=P(Z\le 0.87)=0.8078\)

Using Minitab

To find the area to the left of z = 0.87 in Minitab...

1. From the Minitab menu select Calc> Probability Distributions> Normal.
2. Select Cumulative Probability.
3. In the Input constant box, enter 0.87. Click OK

You should see a value very close to 0.8078.

Based on the definition of the probability density function, we know the area under the whole curve is one. Since we are given the “less than” probabilities in the table, we can use complements to find the “greater than” probabilities. Therefore,

\(P(Z>0.87)=1-P(Z\le 0.87)\).

Using the information from the last example, we have \(P(Z>0.87)=1-P(Z\le 0.87)=1-0.8078=0.1922\)

Using Minitab

Since we are given the “less than” probabilities when using the cumulative probability in Minitab, we can use complements to find the “greater than” probabilities. Therefore,

\(P(Z>0.87)=1-P(Z\le 0.87)\).

Using the information from the last example, we have \(P(Z>0.87)=1-P(Z\le 0.87)=1-0.8078=0.1922\)

You can also use the probability distribution plots in Minitab to find the "greater than."

1. Select Graph> Probability Distribution Plot> View Probability and click OK.
2. In the pop-up window select the Normal distribution with a mean of 0.0 and a standard deviation of 1.0.
3. Select the Shaded Area tab at the top of the window.
4. Select X Value.
5. Enter 0.87 for X value.
6. Select Right Tail.
7. Click OK.

To find the probability between these two values, subtract the probability of less than 2 from the probability of less than 3. In other words,

\(P(2<Z<3)=P(Z<3)-P(Z<2)\)

\(P(Z<3)\)and \(P(Z<2)\)can be found in the table by looking up 2.0 and 3.0.

For 3.0...

For 2.0...

\(P(2 < Z < 3)= P(Z < 3) - P(Z \le 2)= 0.9987 - 0.9772= 0.0215\).

Using Minitab

To find the area between 2.0 and 3.0 we can use the calculation method in the previous examples to find the cumulative probabilities for 2.0 and 3.0 and then subtract.

\(P(2 < Z < 3)= P(Z < 3) - P(Z \le 2)= 0.9987 - 0.9772= 0.0215\)

You can also use the probability distribution plots in Minitab to find the "between."

1. Select Graph> Probability Distribution Plot> View Probability and click OK.
2. In the pop-up window select the Normal distribution with a mean of 0.0 and a standard deviation of 1.0.
3. Select the Shaded Area tab at the top of the window.
4. Select X Value.
5. Select Middle.
6. Enter 2.0 for X value 1 and 3.0 for X value 2.
7. Click OK.

The question is asking for a value to the left of which has an area of 0.1 under the standard normal curve.

Since the entries in the Standard Normal Cumulative Probability Table represent the probabilities and they are four-decimal-place numbers, we shall write 0.1 as 0.1000 to remind ourselves that it corresponds to the inside entry of the table. We search the body of the tables and find that the closest value to 0.1000 is 0.1003. We look to the leftmost of the row and up to the top of the column to find the corresponding z-value.
The corresponding z-value is -1.28. Thus z = -1.28.

Therefore, the 10th percentile of the standard normal distribution is -1.28.

Using Minitab

To find the 10th percentile of the standard normal distribution in Minitab...

1. Select Calc> Probability Distributions> Normal.
2. In the new window choose Inverse Cumulative Probability.
3. Enter 0.1 in the Input constant box.
4. Click OK.

You should see a value very close to -1.28.

• Method 1: Using Minitab
• Method 2: Using a table