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The process for finding the area to the left of a given z-score involves using the standard normal distribution table or a statistical software to determine the probability of obtaining a value less than the given z-score. This area is also known as the cumulative probability or the p-value.
This process is commonly used in statistics to analyze and interpret data. For example, in hypothesis testing, the area to the left of a certain z-score is used to determine the significance of a test statistic. A smaller area indicates a stronger evidence against the null hypothesis, while a larger area suggests weaker evidence.
In addition, the area to the left of a z-score is also used in calculating confidence intervals, where it represents the probability of obtaining a value within a certain range.
Overall, the process of finding the area to the left of a given z-score is a fundamental step in many statistical analyses and helps to provide insights into the likelihood of a particular outcome occurring.
Find Area to the Left of Z-Score (With Examples)
In statistics, a tells us how many standard deviations away a given value lies from a population mean.
We use the following formula to calculate a z-score for a given value:
z = (x – μ) / σ
where:
- x: Individual data value
- μ: Mean of population
- σ: Standard deviation of population
To find the area under a normal distribution that lies to the left of a given z-score, we can use one of two methods:
1. Use the .
2. Use the .
The following examples show how to use each of these methods in practice.
Example 1: Area to the Left of Negative Z-Score
The weight of a certain species of turtles is normally distributed with mean μ = 300 pounds and standard deviation σ = 15 pounds. Approximately what percentage of turtles weigh less than 284 pounds?
The z-score for a weight of 284 pounds would be calculated as z = (284 – 300) / 15 = -1.07
We can use one of two methods to find the area to the left of this z-score:
Method 1: Use z table.
To find the area to the left of the z-score, we can simply look up the value -1.07 in the :
The area to the left of z = -1.07 is 0.1423.
Method 2: Use Area to the Left of Z-Score Calculator
We can also use the to find that the area to the left of z = -1.07 is 0.1423.
Example 2: Area to the Left of Positive Z-Score
The scores on a certain exam are normally distributed with mean μ = 85 and standard deviation σ = 8. Approximately what percentage of students score less than 87 on the exam?
The z-score for an exam score of 87 would be calculated as z = (87 – 85) / 8 = 0.25
We can use one of two methods to find the area to the left of this z-score:
Method 1: Use z table.
To find the area to the left of the z-score, we can simply look up the value 0.25 in the :
The area to the left of z = 0.25 is 0.5987. Applied to our scenario, this means approximately 59.87% of students score less than 87 on this exam.
Method 2: Use Area to the Left of Z-Score Calculator
We can also use the to find that the area to the left of z = 0.25 is 0.5987.
The following tutorials provide additional information on how to work with z-scores:
Cite this article
stats writer (2024). What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?. PSYCHOLOGICAL SCALES. Retrieved from https://scales.arabpsychology.com/stats/what-is-the-process-for-finding-the-area-to-the-left-of-a-given-z-score-and-what-are-some-examples-of-how-it-is-used/
stats writer. "What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?." PSYCHOLOGICAL SCALES, 5 May. 2024, https://scales.arabpsychology.com/stats/what-is-the-process-for-finding-the-area-to-the-left-of-a-given-z-score-and-what-are-some-examples-of-how-it-is-used/.
stats writer. "What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?." PSYCHOLOGICAL SCALES, 2024. https://scales.arabpsychology.com/stats/what-is-the-process-for-finding-the-area-to-the-left-of-a-given-z-score-and-what-are-some-examples-of-how-it-is-used/.
stats writer (2024) 'What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?', PSYCHOLOGICAL SCALES. Available at: https://scales.arabpsychology.com/stats/what-is-the-process-for-finding-the-area-to-the-left-of-a-given-z-score-and-what-are-some-examples-of-how-it-is-used/.
[1] stats writer, "What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?," PSYCHOLOGICAL SCALES, vol. X, no. Y, ص Z-Z, May, 2024.
stats writer. What is the process for finding the area to the left of a given z-score, and what are some examples of how it is used?. PSYCHOLOGICAL SCALES. 2024;vol(issue):pages.