Introduction to Confidence Intervals: Study with Video Lessons, Practice Problems & Examples

Estimating a parameter, such as the mean (μ), often involves using a point estimate (x̄). However, a more informative approach is to construct a confidence interval, which provides a range of values likely to contain the parameter. The confidence level (c), such as 95%, indicates the probability that the interval includes the parameter, with α representing the area outside the interval. The margin of error (e) defines the distance from the point estimate to the interval's endpoints, calculated as ŷ ± e. Critical z values, like z

_α/2, are essential for determining the margin of error.

Introduction to Confidence Intervals

Hey, everyone. Whenever we're faced with estimating a parameter, like, say, mu, we can do so using a single point estimate like we see here. This may be something like, say, x-bar. Now, because our point estimate is rarely going to give us the exact value of the parameter we're trying to estimate, it could actually be more useful to come up with a range of values that our parameter is likely to fall within, rather than just a single point. Now, this range of values is exactly what we're going to talk about here.

This is referred to as a confidence interval. So here, we're gonna talk about what a confidence interval is, and we'll construct and interpret one together. And we'll, of course, look at all of the vocabulary and notation associated with that. So let's go ahead and get started. Now our ultimate goal here is to construct this range of values, this confidence interval, but before we get there, we're going to talk about a couple of other things first.

Let's take a look at part a of our example. Here, we're asked to find alpha for a 95% confidence interval. Now let's ignore alpha for a second. We're going to come right back to it and focus on this 95% confidence interval. Now, this 95% represents what's called our confidence level, and our confidence level is the probability that our confidence interval, that the range of values that we have, actually contains the parameter that we're trying to estimate.

Now, this can be any percentage; in this case, it's 95%, and looking at our curve down here, this then means that when we come up with this range of values with this confidence interval, we can then say that we are 95% confident that this range of values contains our parameter. Now this represents the area underneath my curve within this range of values because this is a probability. Now when working with confidence level, you'll often see it noted with a capital C, which is said to be equal to $1-\alpha$. So this is where alpha comes in. Now depending on your textbook or your professor, you may see notation here just in terms of C or just in terms of alpha, but regardless, we're working with the same calculations and the same values.

So what actually is this alpha? Well, if C is equal to $1-\alpha$ and we know that C is the area underneath my curve here within this range of values, what about this additional area out to the side here? This additional area, the area underneath these tails outside of my confidence interval is alpha. So if C is equal to $1-\alpha$, that means that alpha is equal to $1-C$. It's the complement of C.

It's the probability that our parameter actually falls outside of that interval. So in this case, alpha is equal to $1-$ that confidence level of 95%, so 0.95. This gives us an alpha value of 0.05. Now if this is the area of both of these tails combined, that then means that each of their individual areas is alpha over 2. So in this case, that's $0.05/2$ or 0.025.

Now alpha will become a more important leader as we'll actually end up using it to calculate z-scores associated with confidence intervals. But for now, let's move on to part b of our example. In part b, we're still working with this 95% confidence interval and we want to make a 95% confidence interval for a parameter y, given a point estimate $\hat{y}$ of 4 and a margin of error e of 2. Now what exactly is margin of error? Before we can go about calculating that, we need to know what this actually is.

Now margin of error is the maximum likely amount of error. But for practical purposes here with our confidence interval, it is the distance between whatever our point estimate is and the endpoints of our confidence interval. So coming back down to our curve here, if this is my point estimate, $\hat{y}$, the distance from that point estimate out to the endpoint of my confidence interval represents e, my margin of error, on either side of that point estimate. That means that the endpoints of my confidence interval can be found by taking our point estimate and subtracting and adding our margin of error. So that means that we can expect our parameter y to be in between these two endpoints, $\hat{y}-e$ and $\hat{y}+e$.

Again, that's our point estimate plus and minus our margin of error e. Now here, since we're explicitly given our point estimate and our margin of error, that means that we can then find the endpoints of our confidence interval. So, taking my point estimate here, 4, and subtracting 2, and also taking that point estimate 4 and adding 2, that's my margin of error. This then gives me my 2 endpoints here of 26. So these are the endpoints of my confidence interval.

But how can I actually interpret this result given these endpoints and this confidence level? When we were working with a point estimate, we could say that our parameter that we were working with, so in this case over here that that is ${\mu }_x$, we could say that it was most likely equal to what our point estimate was. So in this case, 4. But when working with a confidence interval, we need to take into account our confidence level as well as the range of values that we're working with. So in order to interpret this result, we would say that we are 95% confident because that's our confidence level, that our parameter y is in between these two values, 26.

That's what our endpoints were found to be. And that's our full confidence interval. Now when working with confidence intervals, you can state your answer in slightly different notations, and you may see these depending on your textbook or your professor. Now the first is interval notation where you just write your 2 endpoints in parentheses. So that's your point estimate, $\hat{y}-e$, and $\hat{y}+e$.

Just your 2 endpoints enclosed in parentheses there. Now you can also write this in its most compact form. That's your point estimate, so $\hat{y}\pm e$. Now these are the absolute basics of confidence intervals. And when constructing this confidence interval, we were explicitly given our margin of error, but this will not always be the case.

And as we continue to work through problems, we're going to have to actually find this margin of error. And as we continue to work through this chapter, I'm gonna give you all the tools you need to solve all of these confidence interval problems. But for now, let's get a bit more practice with the basics of confidence intervals. I'll see you in the next video.

Find $\alpha$ for a 90% confidence interval.

• A. $\alpha=0.90$
• B. $\alpha=0.10$
• C. $\alpha=0.05$
• D. $\alpha=0.01$

Answer: (B)

Make a 90% confidence interval for a parameter, y, with point estimate $ŷ=-1.5$, & margin of error $E=3.25$.

• A. (-1.5, 3.25)
• B. (1.75, 4.75)
• C. (-3.25, -1.5)
• D. (-4.75, 1.75)

Answer: (D)

Critical Values: z Scores

In order to construct a confidence interval, we have to know the margin of error. More often than not, we will not be given this margin of error explicitly, so we'll have to calculate it. We can do this by using a critical value like zα2, which is literally just a z score, something that we found before when working with probabilities in the past. The only thing that's really different here is the notation. So here, I'm going to walk you through how to find a z score associated with a confidence interval so that we can ultimately find the margin of error needed to construct a confidence interval. So let's go ahead and get started.

Now, given our standard normal distribution, our critical z values, this negative zα2 and positive zα2, represent the endpoints of our confidence interval, separating out these tail ends on either side. Now, from working with z scores in the past, we know that we need to use the area to the left of our boundary to ultimately find our z score. So when working with our confidence interval here, we can see that the area to the left of this critical z value is just the area of our tail ends here. Now from working with confidence intervals in the past here, we know that the total area of these two tails together is given by α, which is 1 minus c, our confidence level. Now for this problem, since we're working with a 95% confidence interval, that means that α is then equal to 0.05.

This is the combined area of both of these tail ends. That's α, which is ultimately the probability that our parameter is not contained within that confidence interval. But here, we only want the area of just one of our tails. So, we need to take that and divide it by 2. This gives us 0.025 as the area of just one tail end or the other.

This is ultimately the probability that our parameter is contained in just one of these tails. Now from this value, this 0.025, the area of that tail end, we want to ultimately use that to find our critical z value, which we can do by either using a calculator or by looking this up in a z table. Now, whatever method you choose to use in order to find your critical z value, make sure that you know what area you should be using. If we're looking up negative zα2, we can just use that α/2 area that we just found, that 0.025. But if you're using a calculator or you're looking for positive zα2, remember that our area to the left of this boundary is α/2 plus that entire confidence interval, that 0.95.

So make sure that you're using the correct area in order to find your z value. However, you choose to look this up, calculator or z table, you should ultimately get that zα2 is equal to 1.96, which we can then label on our graph here. This is negative 1.96 and positive 1.96 for our critical z values. Now, in finding this critical z value, the only thing that we used was our 95% confidence level. So what this means is that zα2, our critical z score, will always be the same for any particular confidence level.

So no matter what else is going on in any problem, if I'm working with a 95% confidence interval, my z score will always be 1.960. Now this means that we can then take a look at some critical z scores for some common confidence levels. We already know that for a 95% confidence interval, this results in a critical z score of 1.960. Now, another common confidence level that we'll encounter is 90%. So, we can go ahead and calculate α/2, 1 minus 0.9 over 2, to give us 0.05, which results in a value of zα/2, which you may also see written as z_c.

That's just our critical z value. And we'll ultimately get that this is 1.645, either using a calculator or looking this up in a z table. Now let's look at one other common confidence level, this 99%. We can do the same calculation, 1 minus 0.99, that confidence level, over 2, which gives us 0.005, which results in a critical z value of 2.576. Now you may encounter other confidence levels as you work through problems, but these are three common ones.

So now that we know how to find z scores associated with confidence intervals, let's keep moving down that path to ultimately construct a confidence interval and find that margin of error to do so. Thank you for watching, and I'll see you in the next one.

Find the critical value, $z_{\frac{\alpha }{2}}$, for a 80% confidence interval.

• A. 0.10
• B. 1.282
• C. 0.40
• D. 0.90

Answer: (B)

Here’s what students ask on this topic:

What is a confidence interval in statistics?

A confidence interval in statistics is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. For example, a 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of the intervals to contain the population parameter.

How do you calculate the margin of error for a confidence interval?

The margin of error (e) for a confidence interval can be calculated using the formula:

e=zα/2×σ/n

where zα/2 is the critical z value, σ is the population standard deviation, and n is the sample size. The critical z value corresponds to the desired confidence level (e.g., 1.96 for 95% confidence).

What is the relationship between confidence level and alpha (α) in confidence intervals?

The confidence level (c) and alpha (α) are complementary probabilities in the context of confidence intervals. The confidence level represents the probability that the interval contains the population parameter, while alpha represents the probability that the parameter lies outside the interval. Mathematically, this relationship is expressed as:

c=1-α

For example, a 95% confidence level corresponds to an alpha of 0.05 (5%).

How do you interpret a 95% confidence interval?

To interpret a 95% confidence interval, you would say that you are 95% confident that the true population parameter lies within the interval. For example, if the 95% confidence interval for a population mean is (4, 6), you would state that you are 95% confident that the true mean lies between 4 and 6. This means that if you were to take many samples and construct a confidence interval from each sample, approximately 95% of those intervals would contain the true population mean.

What are critical z values and how are they used in constructing confidence intervals?

Critical z values are the z scores that correspond to the desired confidence level in a standard normal distribution. They are used to determine the margin of error for a confidence interval. For example, for a 95% confidence level, the critical z value is approximately 1.96. This value is used in the formula for the margin of error:

e=zα/2×σ/n

where zα/2 is the critical z value, σ is the population standard deviation, and n is the sample size.