A confidence interval (C.I.) for a difference between means is a range of values that is likely to contain the true difference between two population means with a certain level of confidence. Show
This tutorial explains the following:
C.I. for the Difference Between Means: MotivationOften researchers are interested in estimating the difference between two population means. To estimate this difference, they’ll go out and gather a random sample from each population and calculate the mean for each sample. Then, they can compare the difference between the two means. However, they can’t know for sure if the difference in the sample means matches the true difference in the population means which is why they may create a confidence interval for the difference between the two means. This provides a range of values that is likely to contain the true difference between the population means. For example, suppose we want to estimate the difference in mean weight between two different species of turtles. Since there are thousands of turtles in each population, it would be too time-consuming and costly to go around and weigh each individual turtle. Instead, we might take a simple random sample of 15 turtles from each population and use the mean weight in each sample to estimate the true difference in mean weight between the two populations: The problem is that our samples are random, so the difference in mean weights between the two samples is not guaranteed to exactly match the difference in mean weights between the two populations. So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true difference in mean weight between the two populations. C.I. for the Difference Between Means: FormulaWe use the following formula to calculate a confidence interval for a difference between two means: Confidence interval = (x1–x2) +/- t*√((sp2/n1) + (sp2/n2)) where:
where: C.I. for the Difference Between Means: ExampleSuppose we want to estimate the difference in mean weight between two different species of turtles, so we go out and gather a random sample of 15 turtles from each population. Here is the summary data for each sample: Sample 1:
Sample 2:
Here is how to find various confidence intervals for the true difference in population mean weights: 90% Confidence Interval: (310-300) +/- 1.70*√((305.61/15) + (305.61/15)) = [-0.8589, 20.8589] 95% Confidence Interval: (310-300) +/- 2.05*√((305.61/15) + (305.61/15)) = [-3.0757, 23.0757] 99% Confidence Interval: (310-300) +/- 2.76*√((305.61/15) + (305.61/15)) = [-7.6389, 27.6389] Note: You can also find these confidence intervals by using the Statology Confidence Interval for the Difference Between Means Calculator. You’ll notice that the higher the confidence level, the wider the confidence interval. This should make sense because wider intervals are more likely to contain the true population mean, thus we’re more “confident” that the interval contains the true population mean. C.I. for the Difference Between Means: InterpretationThe way we would interpret a confidence interval is as follows:
Since this interval contains the value “0” it means that it’s possible that there is no difference in the mean weight between the turtles in these two populations. In other words, we cannot say with 95% confidence that there is a difference in mean weight between the turtles in these two populations.
In order to estimate the value of an unknown population mean, we can use a confidence interval based on the standard normal distribution (z-interval) or the t-distribution (t-interval). The choice is usually determined by the following rule:
In both cases, you can either use the formula to compute the interval by hand or use a graphing calculator (or other software). In this article, we will see how to use the TI83/84 calculator to calculate z and t intervals. Note: You can scroll down to see a video of these steps! z-Intervals
In this example, we are told that the population standard deviation is thought to be 14 minutes and we have a large enough sample size. Therefore, a z-interval can be used to calculate the confidence interval. Step 1: Go to the z-interval on the calculator.Press [STAT]->Calc->7. Z-interval [ENTER] Step 2: Highlight STATSSince we have statistics for the sample already calculated, we will highlight STATS at the top. If you had a list of data instead, you could enter it into L1 and then use DATA. This is really only useful for small data sets. Step 3: Enter DataNotice that the calculator asks for sigma (the population standard deviation) – this should help you remember that you shouldn’t be using this unless you have sigma already. Step 4: Calculate and interpretPlease make sure to read the explanation of how to interpret intervals in general. It is very easy to misunderstand what they truly mean! Highlight CALCULATE and then press ENTER to get your interval. Here we are given our 95% confidence interval as (62.149, 71.851). This means that we are 95% confident that the true mean time that college students spend browsing the internet each day is between 62.15 minutes and 71.85 minutes. Note – there are many other ways to write this interval. t-intervalsWhen the population standard deviation is unknown, we will use a t-interval to estimate the mean. Using a t-interval on the calculator has almost the same steps as using a z-interval, as you will see below.
Notice that in this example, the standard deviation is coming from the sample. Therefore, this is not sigma (the population standard deviation) but instead s (the sample standard deviation). This means we should use a t-interval for this estimate. Step 1: Go to the t-interval on the calculator.Press [STAT]->Calc->8. t-interval then [ENTER]. Step 2: Highlight STATSSince we have statistics for the sample already calculated, we will highlight STATS at the top. If you had a list of data instead, you could enter it into L1 and then use DATA. This is really only useful for small data sets. Step 3: Enter DataNotice this time that the calculator asks for s – the sample standard deviation. Step 4: Calculate and interpretAll you have to do is highlight CALCULATE and press ENTER. So our 99% confidence interval is (11.16, 17.24). We can interpret this by saying “We are 99% confident that the mean number of years spent working in education by high school teachers in this community is between 11.16 years and 17.24 years.”Video walkthroughThe following video will walk you through the steps for a t-interval or z-interval on the TI83/84 calculator. |