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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

This article has been viewed 123,014 times.

A confidence interval is an indicator that helps us know the accuracy of a measurement. In addition, the confidence interval also indicates the stability of estimating a value, i.e., by using the confidence interval, you can see how the results of the repeated measurement will deviate from the original estimate. . The following article will help you know how to calculate confidence intervals.

**Record the phenomenon you want to check.**Let’s say you want to test the following scenario:

*The average weight of male students at ABC is 81 kg (equivalent to 180 lbs)*. You need to check that your prediction about the weight of male students in ABC school is correct within the given confidence interval.

**Select a sample from the given population.**This is the step you will take to collect data to test your hypothesis. For example, you have randomly selected 1000 male students.

**Calculate the mean and standard deviation of the sample.**Select a sample statistic (e.g., sample mean, standard deviation) that you want to use to estimate the population parameter you selected. A population parameter is a value that represents a certain characteristic of that population. To calculate the mean and standard deviation of the sample, do the following:

- We calculate the mean by taking the total weight of 1000 selected male students and dividing the total obtained by 1000, i.e. the number of students. The average weight value obtained will be 81 kg (180 lbs).
- To calculate the standard deviation, you need to determine the mean of the data set. Then you need to calculate the variability of the data, or in other words find the mean of the square of deviation from the mean. Next, we will take the square root of the obtained value. Assume the calculated standard deviation is 14 kg (equivalent to 30 lbs). (Note: sometimes the standard deviation will be given in statistical problems.)

**Select the confidence interval you want.**The commonly used confidence intervals are 90%, 95%, and 99%. This value is also usually given. For example, consider a 95% confidence interval.

**Calculate the error range or error limit.**The error limit can be calculated by the formula:

**Z**where Z

_{a/2}* σ/√(n)._{a/2}is the confidence coefficient, where a is the confidence interval, σ is the standard deviation and n is the sample size. In other words, you need to multiply the cut-off value by the standard error. To solve this formula, we divide the formula into small parts as follows:

- To calculate the limit value Z
_{a/2}: The confidence interval under consideration is 95%. Converting from percentage value to decimal value we get: 0.95; divide this value by 2 to get 0.475. Next, compare with the z table to find the value corresponding to 0.475. We see that the closest value is 1.96 at the intersection of row 1.9 and column 0.06. - To calculate the standard error, take the standard deviation of 30 (in lbs, and 14 in kg), divide this value by the square root of the sample size of 1000. We get 30/31.6 = 0.95 lbs, or (14/31.6 = 0.44 kg).
- Multiply the critical value by the standard error, i.e. take 1.96 x 0.95 = 1.86 (in lbs) or 1.96 x 0.44 = 0.86 (in kg). This product is the error limit or range of error.

**Enter the confidence interval.**To record a confidence interval, take the mean (180 lbs, or 81 kg) and write it to the left of the ± sign, then the error limit. So, the result is: 180 ± 1.86 lbs or 81 ± 0.44 kg. The upper and lower bounds of the confidence interval can be determined by adding or subtracting the mean by an amount equal to the error range. That is, in lbs, the lower bound is 180 – 1.86 = 178.16 and the upper bound is 180 + 1.86 = 181.86.

- We can also use this formula to determine the confidence interval:
**x̅ ± Z**where x̅ is the mean value._{a/2}* σ/√(n).

## Advice

- You can calculate t-statistics and z-statistics by hand or use a hand-held calculator with graphs or tables of statistics that are often included in statistics books. The z-statistic can be determined using the Normal Distribution Calculator, while the t-statistic can be calculated using the t-Distribution Calculator. In addition, you can also use support tools available online.
- The sample size needs to be large enough for the confidence interval to be valid.
- The critical value used to calculate the error range is a constant and is expressed as a t-statistic or a z-statistic. The t-statistic is often used when the population standard deviation is unknown or when the sample size is not large enough.
- There are several sampling methods that can help you choose a representative sample for your test, such as simple random sampling, systematic sampling or stratified sampling.
- Confidence intervals do not indicate the probability of a single outcome occurring. For example, with a 95% confidence interval, you can say that the population mean is between 75 and 100. A 95% confidence interval does not mean that you can be 95% certain that the value of the population is between 75 and 100. the average of the test will fall within the range you calculated.

## Things you need

- A set of samples
- Computer
- Network connections
- Statistics textbook
- Handheld computer with graphics

This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.

This article has been viewed 123,014 times.

A confidence interval is an indicator that helps us know the accuracy of a measurement. In addition, the confidence interval also indicates the stability of estimating a value, i.e., by using the confidence interval, you can see how the results of the repeated measurement will deviate from the original estimate. . The following article will help you know how to calculate confidence intervals.

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