The confidence interval is an indicator of the precision of your measurement. It is also an indicator of how stable your estimate is, which is a measure of how close your measurement will be to your original estimate if you repeat the experiment. Follow the steps below to calculate the confidence interval for your data.
Step
Step 1. Write down the phenomenon you want to test
Let's say for example that you are working with the following situation: The average weight of a male student at ABC University is 81.6 kg. You will test how accurately you can predict the weight of male students at ABC University within a certain confidence interval.
Step 2. Select a sample from the population you selected
This is what you will use to collect data for the purpose of testing your hypothesis. Say you have randomly selected 1,000 male students.
Step 3. Calculate the mean and standard deviation of your sample
Select a sample statistic (eg sample mean, sample standard deviation) that you want to use to estimate the selected population parameter. Population parameter is a value that represents a certain population characteristic. Here's how to find the sample mean and sample standard deviation:
- To calculate the mean of the data sample, add the weights of the 1,000 men you selected and divide the result by 1000, the number of men. Then you will get an average weight of 81.6 kg.
- To calculate the sample standard deviation, you must find the mean of the data. Next, you'll need to find the variance of the data, or the average of the sum of the squares of the difference in the data from the mean. Once you find this number, take the root. Let's say the standard deviation here is 13.6 kg. (Note that this information is sometimes given to you while working on statistics problems.)
Step 4. Select the confidence level you want
The most commonly used confidence levels are 90 percent, 95 percent and 99 percent. It may also be provided to you when working on a problem. Let's say you have selected 95%.
Step 5. Calculate your margin of error
You can find the margin of error by using the following formula: Za/2 * /√(n).
Za/2 = confidence coefficient, where a = confidence level, = standard deviation, and n = sample size. There is another way, that is, you have to multiply the critical value by the standard error. Here's how you solve a problem using this formula by breaking it down into sections:
- To determine the critical point, or Za/2: Here, the confidence level is 0, 95%. Convert the percentage to a decimal, 0.95, then divide by 2 to get 0.475. Next, check the z table for a value that corresponds to 0.475. You'll find that the closest point is 1.96, at the intersection between lanes 1, 9 and column 0.06.
- To find the standard error, take the standard deviation, 30, and then divide by the root of the sample size, 1,000. You gain 30/31, 6, or 0.43 kg.
- Multiply 1.96 by 0.95 (your critical point by your standard error) to get 1.86, your margin of error.
Step 6. State your confidence interval
To express a confidence interval, you must take the mean (180), and write it next to the ± and the margin of error. The answer is: 180 ± 1.86. You can find the upper and lower limits of the confidence interval by adding or subtracting the margin of error from the average. So, your lower limit is 180 – 1, 86, or 178, 14, and your upper limit is 180 + 1, 86, or 181, 86.
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You can also use this handy formula to find a confidence interval: x̅ ± Za/2 * /√(n).
Here, x̅ represents the average value.
Tips
- Both the t-value and the z-value can be calculated manually, and you can also use a graphing calculator or statistical table, which is often found in statistics textbooks. The Z value can also be found using the Normal Distribution Calculator, while the t value can be found using the t Distribution Calculator. Online tools are also available.
- Your sample population must be normal for your confidence interval to be valid.
- The critical point used to calculate the margin of error is a constant denoted by a t value or a z value. The t-value is usually preferred where the population standard deviation is unknown or when a small sample is used.
- There are many methods, such as simple random sampling, systematic sampling and stratified sampling, by which you can choose a representative sample with which to test your hypothesis.
- The confidence interval does not indicate the existence of a certain probability of an outcome. For example, if you are 95 percent sure that your population mean is between 75 and 100, then the 95 percent confidence interval does not mean that there is a 95 percent chance that the mean will fall within the calculated range.
