Confidence Intervals - Online Tutor, Practice Problems & Exam Prep

A confidence interval is a statistical range that estimates a parameter, such as the mean, based on sample data. For instance, a 95% confidence interval indicates a 95% certainty that the true mean lies within this range. The formula for a confidence interval is $\mathrm{CI}=\mathrm{x̄}\pm t·s/\sqrt{n}$, where x̄ is the sample mean, t is the t-value from the t-distribution table, s is the standard deviation, and n is the sample size. Understanding degrees of freedom is crucial, calculated as n - 1.

Confidence Interval

Hey, everyone. So in this video, we're going to take a look at confidence intervals. Now, the formal definition of a confidence interval is that it's a specific interval estimate of a parameter determined by using data obtained from a sample. In simplistic terms, a confidence interval is basically us being confident, hence the name "confidence," that the mean lies within a given range. So, for example, if we say we're dealing with a 95% confidence interval, that means we are 95% confident the mean lies within a given interval.

So if we take a look here at the formula for a confidence interval, we're going to say confidence interval is equal to 𝛾 plus or minus tsn. Here, t is the Student's t-statistic, s is our standard deviation, n is our number of measurements, and again, our average or mean is ̅𝛾.

With the confidence interval formula, we have our Student's t statistical table. This is used in our understanding of confidence interval in the comparative data from different experiments. Let's look at the parts of this table to get a better understanding of it. Here, we have first our degrees of freedom, which range from 1 all the way to infinity. Here, infinity means we're dealing with a large amount of measurements, something greater than 120. When we say degrees of freedom, degrees of freedom are equal to your measurements n minus 1. For example, if we have 10 measurements, the degrees of freedom would be 10 minus one, which is equal to 9. So, we'd be right here.

Then we could talk about our confidence level in terms of our measurements, ranging from 50% confidence all the way to 99.9% confident. Of course, we can never be 100% confident because there's going to be a little bit of uncertainty with all of our measurements, but we can get all the way up to 99.9. So, let's say we wanted to look at the 95% confidence interval, we'd look at 95% here and we're at 9 for the degrees of freedom since we're dealing with 10 measurements and they both would meet here. So, our t value would be 2.262.

Just remember, a confidence interval is a way of us giving confidence to our mean or average residing within a particular range, plus or minus.

Confidence Interval

Here in this example question, it says, construct a 95% confidence interval for an experiment that found the mean temperature for a given city in July at 103.5 degrees Celsius with a standard deviation of 1.8 from 10 measurements. So remember our confidence interval, which we will abbreviate as CI, equals our mean or average, plus or minus the value times our standard deviation divided by the square root of n. Here they're telling us the mean temperature is 103.5 degrees Celsius, and this is going to be plus or minus.

Go up to the t-table, and we know that our degrees of freedom are just the number of measurements minus 1. So, we have 10 measurements minus 1, which means we have 9 degrees of freedom, which I'll abbreviate as DOF. So, line up 95% with our 9 degrees of freedom; that'll give us our t-value, which comes out to be 2.262. Our standard deviation is 1.8 divided by the number of measurements square-rooted. So here, when we do that, that's going to be: 103.5 ± 1.28 degrees Celsius.

So that means that our temperature on this day would range between 103.5 C minus 1.28 degrees Celsius to 103.5 degrees Celsius plus 1.28 degrees Celsius. So what does that come out to? Well, that comes out to the range on that day as being as low as 102.22 degrees Celsius to 104.78 degrees Celsius.

Alright. So that's what we would say. We're 95% confident that the range of the temperature on that day lies within this interval. So 102.22 degrees Celsius to 104.78 degrees Celsius with the mean temperature being 103.5 degrees Celsius.

Confidence Intervals Calculations 1

Continuing our discussion on confidence intervals, let's take a look at the following example. Here it says the barium content of a metal ore was analyzed several times by a percent composition process. Here we are given 4 measurements of 0.010, 0.011, 0.004, and 0.011. Here, we need to calculate the mean, median, and mode. Now realize here that we have 2 types of means that we've dealt with up to this point. We have our traditional mean, which is just x with a line above it and then we have μ which is termed our population mean. Remember, to get the population mean, we would need to do several measurements, so many in fact that our typical average mean would transition into the population mean. Since here we only have 4 measurements, we know we're dealing with our standard mean. So we're going to say here our mean equals all the measurements added up. So we're going to take all of these measurements, add them together and then divide it by the total number of measurements. So the total number of measurements is 4. Doing that, we're going to have 0.036 as the total for the top portion divided by 4. So that gives me 0.009 as my standard mean.

Next, we need to figure out our median. For our median, we're going to list these measurements from smallest to largest. So we have 0.004, 0.010, 0.011 and 0.011. For the median, we look for the middle of all these measurements. If it's just one number or if it were 5 digits, there would be one number in the middle and we would choose that number as our median. In this example though, two numbers are technically in the middle. So what we do here is we take those two numbers and we add them together and we divide by 2. Doing that would give me 0.021 divided by 2, which is 0.0105 as my median.

Finally, we need to determine our mode. Our mode is just the measurement that appears the most. The measurement that appears the most is 0.011. So that will represent our mode. So these are just some basic, statistical definitions and calculations that you should be familiar with. Now that we've done this example, continue on to the next one where you're asked to calculate the standard deviation. Attempt it on your own, but if you get stuck, just come back and see how I approach that same question.

Confidence Intervals Calculations 1

So from example 1, they asked us to calculate the standard deviation. So remember, your standard deviation, which is \( s \), equals the square root of the summation of each measurement minus the average or mean, and that will be squared, divided by \( n \), which is the number of measurements, minus 1. Now remember, we're dealing with only 4 measurements here so we're just dealing with our standard deviation. If we were dealing with numerous measurements, then it would change into the population standard deviation, and in that case, it would be called sigma. But here we're just dealing with standard again because there are only 4 measurements. So, the formula for standard deviation is going to be each measurement. Remember, \( X_i \) stands for each measurement, so each one of these minus your mean or average, which was this, and that'll be squared.

All right. So plugging this in, we're going to say we have 0.010 for our first measurement minus the average. And plus 0.004 minus the average. Plus 0.004 minus the average, that's squared. Plus 0.011 minus the average and that’s squared, divided by the number of measurements, which is 4 minus 1. Remember, \( n - 1 \) is also coined the degrees of freedom. So when we subtract and then square all of those and then add them together, that's going to give me square root of 0.00034 divided by 3, which is square root of 0.000011, which gives me 0.003367 at the end for my standard deviation. Remember, the smaller your standard deviation, the more precise your numbers are, the more closely they are to one another. But a small standard deviation does not mean that your measurements are accurate. We can only tell if they're accurate if we compare them to the true value of the barium content within the metal ore.

All right. So again, all we know for sure is that this standard deviation is pretty small, so it's going to be pretty precise in terms of how close the measurements are to one another. Now that we've seen this, attempt to figure out the 95% and 90% confidence intervals; that's left as the practice question. Once you're done with that, come back and take a look and see how I approach that same question.

From the examples given above, find the 90% confidence interval. 

• A. 0.009 ± 0.002758
• B. 0.009 ± 0.003589
• C. 0.009 ± 0.003961
• D. 0.009 ± 0.002581

Here’s what students ask on this topic:

What is a confidence interval and how is it used in statistics?

A confidence interval is a range of values, derived from sample data, that is used to estimate an unknown population parameter. For example, a 95% confidence interval for a mean suggests that we are 95% confident that the true mean lies within this range. The formula for a confidence interval is:

$\mathrm{CI}=\overline{x}\pm t·\frac{s}{\sqrt{n}}$

where $\overline{x}$ is the sample mean, $t$ is the t-value from the t-distribution table, $s$ is the standard deviation, and $n$ is the sample size. Confidence intervals are crucial in inferential statistics as they provide a range within which we expect the true parameter to lie, giving us a measure of reliability for our estimates.

How do you calculate the degrees of freedom for a confidence interval?

The degrees of freedom (df) for a confidence interval are calculated as the sample size minus one. This is represented by the formula:

$\mathrm{df}=n-1$

where $n$ is the number of measurements or sample size. For example, if you have 10 measurements, the degrees of freedom would be:

$10-1=9$

Degrees of freedom are important because they are used to determine the appropriate t-value from the t-distribution table, which is then used in the confidence interval formula.

What is the formula for a confidence interval and what do each of the components represent?

The formula for a confidence interval is:

$\mathrm{CI}=\overline{x}\pm t·\frac{s}{\sqrt{n}}$

where:

• $\overline{x}$ is the sample mean, representing the average of the sample data.
• $t$ is the t-value from the t-distribution table, which depends on the desired confidence level and degrees of freedom.
• $s$ is the standard deviation of the sample, indicating the spread of the data.
• $n$ is the sample size, or the number of measurements.

This formula helps estimate the range within which the true population parameter is likely to fall, providing a measure of confidence in the estimate.

What does a 95% confidence interval mean?

A 95% confidence interval means that we are 95% confident that the true population parameter, such as the mean, lies within the calculated interval. This does not mean that 95% of the data falls within this range, but rather that if we were to take many samples and calculate a confidence interval for each, 95% of those intervals would contain the true mean. For example, if the 95% confidence interval for a sample mean is (10, 20), we are 95% confident that the true population mean lies between 10 and 20. This level of confidence is commonly used in statistics to provide a balance between precision and reliability.

How do you interpret the t-value in the confidence interval formula?

The t-value in the confidence interval formula is a critical value from the t-distribution table that corresponds to the desired confidence level and the degrees of freedom. It adjusts the width of the confidence interval based on the sample size and variability. For example, for a 95% confidence level with 9 degrees of freedom, the t-value is approximately 2.262. This value is multiplied by the standard deviation divided by the square root of the sample size to determine the margin of error. The larger the t-value, the wider the confidence interval, reflecting greater uncertainty in the estimate. The t-value accounts for the fact that we are using a sample to estimate the population parameter.