2. what does a confidence interval represent?
A confidence interval does not quantify variabilityA 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. This is not the same as a range that contains 95% of the values. The graph below emphasizes this distinction. Show
 The graph shows three samples (of different size) all sampled from the same population. With the small sample on the left, the 95% confidence interval is similar to the range of the data. But only a tiny fraction of the values in the large sample on the right lie within the confidence interval. This makes sense. The 95% confidence interval defines a range of values that you can be 95% certain contains the population mean. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.
A 95% chance of what?It is correct to say that there is a 95% chance that the confidence interval you calculated contains the true population mean. It is not quite correct to say that there is a 95% chance that the population mean lies within the interval. What's the difference? The population mean has one value. You don't know what it is (unless you are doing simulations) but it has one value. If you repeated the experiment, that value wouldn't change (and you still wouldn't know what it is). Therefore it isn't strictly correct to ask about the probability that the population mean lies within a certain range. In contrast, the confidence interval you compute depends on the data you happened to collect. If you repeated the experiment, your confidence interval would almost certainly be different. So it is OK to ask about the probability that the interval contains the population mean. It is not quite correct to ask about the probability that the population mean is within the interval. It either is in the interval or it isn't. There is no chance about it. What you can say is that if you perform this kind of experiment many times, the confidence intervals would not all be the same, you would expect 95% of them to contain the population mean, you would expect 5% of the confidence intervals to not include the population mean, and you would never know whether the interval from a particular experiment contained the population mean or not. Nothing special about 95%While confidence intervals are usually expressed with 95% confidence, this is just a tradition. Confidence intervals can be computed for any desired degree of confidence. People are often surprised to learn that 99% confidence intervals are wider than 95% intervals, and 90% intervals are narrower. But this makes perfect sense. If you want more confidence that an interval contains the true parameter, then the intervals will be wider. If you want to be 100.000% sure that an interval contains the true population, it has to contain every possible value so be very wide. If you are willing to be only 50% sure that an interval contains the true value, then it can be much narrower.
By Dr. Saul McLeod, published June 10, 2019, updated 2021
What does a 95% confidence interval mean?The 95% confidence interval is a range of values that you can be 95% confident contains the true mean of the population. Due to natural sampling variability, the sample mean (center of the CI) will vary from sample to sample. The confidence is in the method, not in a particular CI. If we repeated the sampling method many times, approximately 95% of the intervals constructed would capture the true population mean. Therefore, as the sample size increases, the range of interval values will narrow, meaning that you know that mean with much more accuracy compared with a smaller sample. We can visualize this using a normal distribution (see the below graph). For example, the probability of the population mean value being between -1.96 and +1.96 standard deviations (z-scores) from the sample mean is 95%. Accordingly, there is a 5% chance that the population mean lies outside of the upper and lower confidence interval (as illustrated by the 2.5% of outliers on either side of the 1.96 z-scores). Why do researchers use confidence intervals?It is more or less impossible to study every single person in a population so researchers select a sample or sub-group of the population. This means that the researcher can only estimate the parameters (i.e. characteristics) of a population, the estimated range being calculated from a given set of sample data. Therefore, a confidence interval is simply a way to measure how well your sample represents the population you are studying. The probability that the confidence interval includes the true mean value within a population is called the confidence level of the CI. You can calculate a CI for any confidence level you like, but the most commonly used value is 95%. A 95% confidence interval is a range of values (upper and lower) that you can be 95% certain contains the true mean of the population. How do I calculate a confidence interval?To calculate the confidence interval, start by computing the mean and standard error of the sample. Remember, you must calculate an upper and low score for the confidence interval using the z-score for the chosen confidence level (see table below).
Where:
For the lower interval score divide the standard deviation by the square root on n, and then multiply the sum of this calculation by the z-score (1.96 for 95%). Finally, subtract the value of this calculation from the sample mean. An Example
Lower Value: 86 - 1.960 × 6.2 √46 = 86 - 1.79 = 84.21 Upper Value: 86 + 1.960 × 6.2 √46 = 86 + 1.79 = 87.79 So the population mean is likely to be between 84.21 and 87.79 How can we be confident the population mean is similar to the sample mean?The narrower the interval (upper and lower values), the more precise is our estimate. As a general rule, as a sample size increases the confident interval should become more narrow. Therefore, with large samples, you can estimate the population mean with more precision than you can with smaller samples, so the confidence interval is quite narrow when computed from a large sample. How to report a confident interval APA styleThe APA 6 style manual states (p.117): “ When reporting confidence intervals, use the format 95% CI [LL, UL] where LL is the lower limit of the confidence interval and UL is the upper limit. ” For example, one might report: 95% CI [5.62, 8.31]. Confidence intervals can also be reported in a table How to reference this article:How to reference this article:McLeod, S. A. (2019, June 10). What are confidence intervals in statistics? Simply psychology: https://www.simplypsychology.org/confidence-interval.html Home | About Us | Privacy Policy | Advertise | Contact Us Simply Psychology's content is for informational and educational purposes only. Our website is not intended to be a substitute for professional medical advice, diagnosis, or treatment. © Simply Scholar Ltd - All rights reserved What do Confidence intervals represent?Confidence intervals are one way to represent how "good" an estimate is; the larger a 90% confidence interval for a particular estimate, the more caution is required when using the estimate. Confidence intervals are an important reminder of the limitations of the estimates.
What does a 95% confidence interval tell you?The 95% confidence interval defines a range of values that you can be 95% certain contains the population mean. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.
What does a confidence interval represent quizlet?What is a confidence interval? A confidence interval measures the probability that a population parameter will fall between two set values. A confidence interval is the probability that a value will fall between an upper and lower bound of a probability distribution.
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