A confidence interval is an indicator of the accuracy of measurements. It is also an indicator of how stable an estimate is, measuring how close your measure is to the original estimate if you repeat your experiment. Follow the steps below to calculate the confidence interval for your data.
Steps
Step 1. Write down the phenomenon you would like to test
Suppose you are working with the following situation. "The average weight of a male student at ABC University is 180 pounds." You will test how accurately you are able to predict the weight of an ABC University male student within a given confidence interval.
Step 2. Select an example from the chosen population
This is what you will use to collect data to test your hypotheses. Let's say you have randomly selected 1000 students.
Step 3. Calculate your sample mean and standard deviation
Choose a reference statistic (e.g. mean, standard deviation) that you want to use to estimate the parameter on the chosen population. A population parameter is a value that represents a particular characteristic of the population. You can find the mean and standard deviation as follows:
- To calculate the sample mean, add all the weights of the 1000 men you selected and divide the result by 1000, the number of men. This should give you an average of 186 lbs.
- To calculate the sample standard deviation, you will need to find the mean, or mean, of the data. Next, you will need to find the variance of the data, or the mean of the differences from the mean squared. Once you have found these numbers, just take the square root. Let's say the standard deviation is 30 pounds (note that this information can sometimes be given to you in a statistical problem).
Step 4. Choose the desired confidence interval
The most used confidence intervals are those of 90, 95 and 99%. This can also be indicated to you within a problem. Let's say you chose 95%.
Step 5. Calculate your margin of error
You can find the margin of error using the formula: Za / 2 * σ / √ (n).
Za / 2 = confidence coefficient, where a = confidence level, σ = standard deviation, and n = sample size. This is another way of saying that you need to multiply the critical value by the standard error. Here's how you can solve this formula by breaking it into parts:
- To find the critical value, or Za / 2: here the confidence level is 95%. Convert the percentage to decimal, 0, 95, and divide by 2 resulting in 0, 475. So, check the z table to find the value corresponding to 0, 475. You will see that the closest value is 1. 96, at the intersection of the row 1, 9 and column 0, 06.
- Take the standard error, and the standard deviation, 30, and divide by the square root of the sample size, 1000. You'll get 30/31, 6, or.95 lbs.
- Multiply 1.95 by 0.95 (your critical value given by the standard error) to get 1.86, your margin of error.
Step 6. Set your confidence interval
To set the confidence interval, you have to take the mean (180), and write it with ± and then the margin of error. The answer is: 180 ± 1.86. You can find the upper and lower bounds of the confidence interval by adding and subtracting the margin of error from the mean. 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 the confidence interval: x̅ ± Za / 2 * σ / √ (n).
. Here, x̅ represents the mean.
Advice
- Both t and z can be calculated manually, for example using a graphing calculator or statistical tables, which are often found in statistics books. Z can be found using the normal distribution calculator, while t can be found with the distribution calculator. Online tools are also available.
- The critical value used to calculate the margin of error is a constant which is expressed as a t or a z. T's are usually preferred when the population standard deviation is not known or when a small sample is used.
- Your sample population must be normal for your confidence interval to be valid.
- A confidence interval does not indicate the likelihood of a particular outcome occurring. For example, if you are 95% sure that your population mean is between 75 and 100, the 95% confidence interval does not mean that there is a 95% probability that the mean falls within the range you calculated..
- There are many methods, such as simple random sampling, systematic sampling, and stratified sampling, from which you can select a representative sample that you can use to test your hypothesis.
