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The Z table (also known as the standard normal table) is a mathematical table that allows us to know the percentage of values below a particular z-score in a standard normal distribution. It is used to find probabilities associated with the normal distribution, and it is typically used in hypothesis testing, confidence intervals, and other statistical tests. To use the Z table, you must first find the corresponding z-score that corresponds to the area of the normal distribution you are looking for, then look up the area in the Z table to find the probability. Examples of how to use the Z table are provided in most textbooks.
A is a table that tells you what percentage of values fall below a certain z-score in a standard normal distribution.
A z-score simply tells you how many standard deviations away an individual data value falls from the mean. It is calculated as:
z-score = (x – μ) / σ
where:
- x: individual data value
- μ: population mean
- σ: population standard deviation
This tutorial shows several examples of how to use the z table.
Example 1
The scores on a certain college entrance exam are normally distributed with mean μ = 82 and standard deviation σ = 8. Approximately what percentage of students score less than 84 on the exam?
Step 1: Find the z-score.
First, we will find the z-score associated with an exam score of 84:
z-score = (x – μ) / σ = (84 – 82) / 8 = 2 / 8 = 0.25
Step 2: Use the z-table to find the percentage that corresponds to the z-score.
Next, we will look up the value 0.25 in the :
Approximately 59.87% of students score less than 84 on this exam.
Example 2
The height of plants in a certain garden are normally distributed with a mean of μ = 26.5 inches and a standard deviation of σ = 2.5 inches. Approximately what percentage of plants are greater than 26 inches tall?
First, we will find the z-score associated with a height of 26 inches.
z-score = (x – μ) / σ = (26 – 26.5) / 2.5 = -0.5 / 2.5 = -0.2
Step 2: Use the z-table to find the percentage that corresponds to the z-score.
Next, we will look up the value -0.2 in the :
We see that 42.07% of values fall below a z-score of -0.2. However, in this example we want to know what percentage of values are greater than -0.2, which we can find by using the formula 100% – 42.07% = 57.93%.
Thus, aproximately 59.87% of the plants in this garden are greater than 26 inches tall.
Example 3
The weight of a certain species of dolphin is normally distributed with a mean of μ = 400 pounds and a standard deviation of σ = 25 pounds. Approximately what percentage of dolphins weigh between 410 and 425 pounds?
Step 1: Find the z-scores.
First, we will find the z-scores associated with 410 pounds and 425 pounds
z-score of 410 = (x – μ) / σ = (410 – 400) / 25 = 10 / 25 = 0.4
z-score of 425 = (x – μ) / σ = (425 – 400) / 25 = 25 / 25 = 1
Step 2: Use the z-table to find the percentages that corresponds to each z-score.
First, we will look up the value 0.4 in the :
Then, we will look up the value 1 in the :
Lastly, we will subtract the smaller value from the larger value: 0.8413 – 0.6554 = 0.1859.
Thus, approximately 18.59% of dolphins weigh between 410 and 425 pounds.