Online Calculator 95 Confidence Interval
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Reviewed by Calculator Editorial Team
A 95% confidence interval is a range of values that is likely to contain the true population parameter with 95% probability. This calculator helps you compute confidence intervals for population means when the population standard deviation is known.
What is a 95% Confidence Interval?
In statistics, a confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. A 95% confidence interval means that if we were to take many samples and compute a 95% confidence interval for each, approximately 95% of these intervals would contain the true population parameter.
Formula
The formula for a 95% confidence interval for a population mean (μ) when the population standard deviation (σ) is known is:
CI = x̄ ± z*(σ/√n)
Where:
- x̄ = sample mean
- z = z-score for 95% confidence (approximately 1.96)
- σ = population standard deviation
- n = sample size
The confidence interval provides a range of plausible values for the population parameter. For example, if you calculate a 95% confidence interval for the average height of adults in a city, you can be 95% confident that the true average height falls within that range.
How to Use This Calculator
Using this calculator is straightforward. Follow these steps:
- Enter the sample mean (x̄) in the first field.
- Enter the population standard deviation (σ) in the second field.
- Enter the sample size (n) in the third field.
- Click the "Calculate" button to compute the confidence interval.
- Review the results and interpretation.
Note: This calculator uses the z-score for 95% confidence (approximately 1.96). If you need a different confidence level, you would use a different z-score.
Interpreting Results
The confidence interval provides a range of values that is likely to contain the true population parameter. For example, if you calculate a 95% confidence interval for the average test score of students in a school, you can interpret the result as follows:
If the confidence interval is (75, 85), you can be 95% confident that the true average test score of all students in the school falls between 75 and 85.
It's important to note that a 95% confidence interval does not mean that there is a 95% probability that the true parameter lies within the interval. Instead, it means that if you were to take many samples and compute a 95% confidence interval for each, approximately 95% of these intervals would contain the true parameter.
Worked Examples
Let's look at a couple of examples to illustrate how to use this calculator.
Example 1: Average Height of Adults
Suppose you want to estimate the average height of adults in a city. You take a random sample of 100 adults and find that the average height is 170 cm with a standard deviation of 5 cm. You want to compute a 95% confidence interval for the average height of all adults in the city.
Using the calculator:
- Sample mean (x̄) = 170
- Population standard deviation (σ) = 5
- Sample size (n) = 100
The calculator would compute the confidence interval as (169.04, 170.96). You can be 95% confident that the true average height of all adults in the city falls between 169.04 cm and 170.96 cm.
Example 2: Average Test Score
Suppose you want to estimate the average test score of students in a school. You take a random sample of 50 students and find that the average test score is 80 with a standard deviation of 10. You want to compute a 95% confidence interval for the average test score of all students in the school.
Using the calculator:
- Sample mean (x̄) = 80
- Population standard deviation (σ) = 10
- Sample size (n) = 50
The calculator would compute the confidence interval as (77.96, 82.04). You can be 95% confident that the true average test score of all students in the school falls between 77.96 and 82.04.
FAQ
What is the difference between a confidence interval and a confidence level?
A confidence level is the percentage that the confidence interval is likely to contain the true population parameter. A 95% confidence level means that if you were to take many samples and compute a 95% confidence interval for each, approximately 95% of these intervals would contain the true parameter.
How do I know if my sample size is large enough for a confidence interval?
The sample size required for a confidence interval depends on the desired margin of error and the variability of the data. A larger sample size will result in a narrower confidence interval, which is more precise. You can use a sample size calculator to determine the appropriate sample size for your study.
What does it mean if the confidence interval includes zero?
If the confidence interval includes zero, it means that the true population parameter could be zero or could be positive or negative. This suggests that there is no statistically significant difference from zero at the specified confidence level.