Throughout this chapter, we've been taking a look at confidence intervals, and we learned how to find their associated critical z-scores. But we have yet to fully construct a confidence interval for a population mean, mu. So that's exactly what we're going to look at here, putting everything that we've learned so far together in a step-by-step process to ultimately construct a confidence interval for you. So let's go ahead and get started here by jumping right into our example. Here, we're told that over 36 trips to work, you find a sample mean of 1 hour.
Now we're asked here to construct a 90% confidence interval for the true population mean travel time, and we're told here to use a population standard deviation of 18 minutes. Now, since we're explicitly given our population standard deviation sigma in this problem, that then means that we can use x bar, our sample mean, as a point estimate and calculate our margin of error using this equation. That e is equal to our critical z-score, that's zα2, times sigma, our population standard deviation, over the square root of n, our sample size. So we can ultimately use this to construct our confidence interval. But before we get there, we have to start from the beginning with step 1.
We first need to verify that we can even follow this process. The first thing that we need to check for here is that our sample is random. Now in most of these problems, you can assume that your sample is random unless you're told that it's not explicitly. In this problem, all we're told is that this is over 36 trips to work. We have no reason to think that the sample isn't random.
So we can go ahead and check that box off. Now the second thing that we want to verify is that x is normal or that our sample size n is greater than 30. This is ultimately to check that this particular random variable has a standard normal distribution. Now, here, we're told that our sample size n is 36. Since this value is greater than 30, we can check that box off and move on to step 2, finding our critical value, zα2.
In order to find zα2, we first need to figure out what alpha over 2 is. Remember that alpha is equal to 1 minus c, our confidence level. So we're just taking that and dividing it by 2. Now in this problem, we have a 90% confidence level. So this becomes 1 minus 0.9 over 2, which gives us a value here of 0.05.
Now from here, you can either use a calculator or a z-table to look up what our critical z-value is with this alpha over 2 value of 0.05. Whatever way you choose to do this, you should end up finding that z alpha over 2 is equal to 1.645 for this 90% confidence level. So with step 2 done, we can move on to step 3 and actually find our margin of error. This is using the equation that we just learned up there. This is z alpha over 2, our critical z-value, times sigma over the square root of n.
So let's go ahead and plug all of those values in to find our margin of error. First, taking that critical z-value we just found, 1.645, then we're multiplying that by that population standard deviation 18 over the square root of our sample size and that is 36. Now when we multiply this out, we will ultimately get that our margin of error here is equal to 4.935. So that's step 3 done. Moving on to our final step here where we're actually going to construct that confidence interval.
We're going to take our sample mean here, x bar, and subtract our margin of error for that lower bound and add our margin of error for that upper bound. So constructing our confidence interval here, abbreviating this CI for confidence intervals, I'm going to take my sample mean of 1 hour. Remember, this is equivalent to 60 minutes. So taking that 60 minutes and then subtracting my margin of error, 4.935, and then adding 4.935 to 60 for that upper bound. Subtracting and adding here, this ends up giving me 55.065 for that lower bound and 64.935 for that upper bound.
So what does this actually mean? Well, given our confidence level of 90% and this range of values from 55.065 to 64.935, we can then interpret this as saying that we are 90% confident, based on that confidence level, that the mean travel time to work is between these two values, 55.065 minutes and 64.935 minutes. So that is our fully constructed and interpreted confidence interval. Now, we're going to get some more practice coming up next. If you have any questions, feel free to let us know.
I'll see you in the next video.
