Find The Area Under The Standard Normal Curve To The Left Of $z = -2.28$. Round Your Answer To 4 Decimal Places.
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Introduction
The standard normal curve, also known as the z-distribution, is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is a fundamental concept in statistics and is used to model many real-world phenomena. In this article, we will discuss how to find the area under the standard normal curve to the left of a given z-score.
Understanding the Standard Normal Curve
The standard normal curve is a continuous probability distribution that is defined by the following probability density function:
f(z) = (1/√(2π)) * e(-z2/2)
where z is the z-score, and e is the base of the natural logarithm.
The standard normal curve is symmetric about the mean of 0, and its standard deviation is 1. This means that the curve is bell-shaped, with the majority of the data points concentrated around the mean.
Finding the Area Under the Standard Normal Curve
To find the area under the standard normal curve to the left of a given z-score, we can use a standard normal distribution table, also known as a z-table. The z-table is a table that lists the area under the standard normal curve to the left of a given z-score.
Using a Standard Normal Distribution Table
A standard normal distribution table is a table that lists the area under the standard normal curve to the left of a given z-score. The table is typically organized in the following way:
| z-score | Area to the left |
|---|---|
| -3.09 | 0.0009 |
| -3.00 | 0.0013 |
| -2.90 | 0.0018 |
| ... | ... |
| 2.90 | 0.9982 |
| 3.00 | 0.9987 |
| 3.09 | 0.9991 |
To find the area under the standard normal curve to the left of a given z-score, we can look up the z-score in the table and read off the corresponding area.
Finding the Area Using a Calculator
Alternatively, we can use a calculator to find the area under the standard normal curve to the left of a given z-score. Most calculators have a built-in function for calculating the area under the standard normal curve.
Finding the Area Under the Standard Normal Curve to the Left of
To find the area under the standard normal curve to the left of , we can use a standard normal distribution table or a calculator.
Using a Standard Normal Distribution Table
Using a standard normal distribution table, we can look up the z-score of -2.28 and read off the corresponding area. The table shows that the area under the standard normal curve to the left of is approximately 0.0111.
Finding the Area Using a Calculator
Alternatively, we can use a calculator to find the area under the standard normal curve to the left of . Most calculators have a built-in function for calculating the area under the standard normal curve.
Conclusion
In this article, we discussed how to find the area under the standard normal curve to the left of a given z-score. We used a standard normal distribution table and a calculator to find the area under the standard normal curve to the left of . The area under the standard normal curve to the left of is approximately 0.0111.
References
- Z-table: A standard normal distribution table that lists the area under the standard normal curve to the left of a given z-score.
- Calculator: A calculator that has a built-in function for calculating the area under the standard normal curve.
Glossary
- Standard normal curve: A probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1.
- Z-score: A measure of how many standard deviations an observation is away from the mean.
- Area under the standard normal curve: The area under the standard normal curve to the left of a given z-score.
Further Reading
- Standard Normal Distribution: A probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1.
- Z-distribution: A probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1.
- Probability density function: A function that describes the probability distribution of a random variable.
Related Articles
- Finding the Area Under the Standard Normal Curve to the Right of
- Finding the Area Under the Standard Normal Curve to the Left of
- Finding the Area Under the Standard Normal Curve to the Right of
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Q: What is the standard normal curve?
A: The standard normal curve, also known as the z-distribution, is a probability distribution that is symmetric about the mean of 0 and has a standard deviation of 1. It is a fundamental concept in statistics and is used to model many real-world phenomena.
Q: What is the z-score?
A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated by subtracting the mean from the observation and dividing by the standard deviation.
Q: How do I find the area under the standard normal curve to the left of a given z-score?
A: You can use a standard normal distribution table or a calculator to find the area under the standard normal curve to the left of a given z-score. Most calculators have a built-in function for calculating the area under the standard normal curve.
Q: What is the difference between the standard normal curve and the normal curve?
A: The standard normal curve and the normal curve are both probability distributions, but they have different means and standard deviations. The standard normal curve has a mean of 0 and a standard deviation of 1, while the normal curve has a mean of μ and a standard deviation of σ.
Q: How do I use a standard normal distribution table?
A: To use a standard normal distribution table, you need to look up the z-score in the table and read off the corresponding area. The table is typically organized in the following way:
| z-score | Area to the left |
|---|---|
| -3.09 | 0.0009 |
| -3.00 | 0.0013 |
| -2.90 | 0.0018 |
| ... | ... |
| 2.90 | 0.9982 |
| 3.00 | 0.9987 |
| 3.09 | 0.9991 |
Q: Can I use a calculator to find the area under the standard normal curve?
A: Yes, you can use a calculator to find the area under the standard normal curve. Most calculators have a built-in function for calculating the area under the standard normal curve.
Q: What is the significance of the standard normal curve in statistics?
A: The standard normal curve is a fundamental concept in statistics and is used to model many real-world phenomena. It is used to calculate probabilities, test hypotheses, and make inferences about populations.
Q: Can I use the standard normal curve to model real-world phenomena?
A: Yes, you can use the standard normal curve to model real-world phenomena. The standard normal curve is a continuous probability distribution that can be used to model many types of data, including heights, weights, and IQ scores.
Q: What are some common applications of the standard normal curve?
A: Some common applications of the standard normal curve include:
- Calculating probabilities: The standard normal curve can be used to calculate probabilities of events occurring.
- Testing hypotheses: The standard normal curve can be used to test hypotheses about populations.
- Making inferences: The standard normal curve can be used to make inferences about populations.
Q: Can I use the standard normal curve to model non-normal data?
A: No, you cannot use the standard normal curve to model non-normal data. The standard normal curve is a continuous probability distribution that assumes normality. If your data is not normally distributed, you may need to use a different distribution, such as the t-distribution or the F-distribution.
Q: What are some common mistakes to avoid when using the standard normal curve?
A: Some common mistakes to avoid when using the standard normal curve include:
- Assuming normality: The standard normal curve assumes normality, so if your data is not normally distributed, you may need to use a different distribution.
- Using the wrong z-score: Make sure to use the correct z-score when using the standard normal curve.
- Not accounting for outliers: The standard normal curve assumes that the data is normally distributed, so if your data has outliers, you may need to use a different distribution.
Q: Can I use the standard normal curve to model categorical data?
A: No, you cannot use the standard normal curve to model categorical data. The standard normal curve is a continuous probability distribution that assumes normality, so it is not suitable for modeling categorical data.
Q: What are some common applications of the standard normal curve in real-world scenarios?
A: Some common applications of the standard normal curve in real-world scenarios include:
- Insurance: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a person dying within a certain age range.
- Finance: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a stock price increasing or decreasing.
- Medicine: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a patient responding to a certain treatment.
Q: Can I use the standard normal curve to model time-series data?
A: No, you cannot use the standard normal curve to model time-series data. The standard normal curve is a continuous probability distribution that assumes normality, so it is not suitable for modeling time-series data.
Q: What are some common mistakes to avoid when using the standard normal curve in real-world scenarios?
A: Some common mistakes to avoid when using the standard normal curve in real-world scenarios include:
- Assuming normality: The standard normal curve assumes normality, so if your data is not normally distributed, you may need to use a different distribution.
- Using the wrong z-score: Make sure to use the correct z-score when using the standard normal curve.
- Not accounting for outliers: The standard normal curve assumes that the data is normally distributed, so if your data has outliers, you may need to use a different distribution.
Q: Can I use the standard normal curve to model complex data?
A: No, you cannot use the standard normal curve to model complex data. The standard normal curve is a continuous probability distribution that assumes normality, so it is not suitable for modeling complex data.
Q: What are some common applications of the standard normal curve in complex data scenarios?
A: Some common applications of the standard normal curve in complex data scenarios include:
- Machine learning: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a machine learning model making a correct prediction.
- Data mining: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a data mining algorithm finding a pattern in the data.
- Computer vision: The standard normal curve can be used to calculate probabilities of events occurring, such as the probability of a computer vision algorithm detecting an object in an image.