Calculating A 99% Confidence Interval For Population Mean With Unknown Variance

Hey guys! Let's dive into a common statistical problem: determining a confidence interval for the population mean (μ) when we don't know the population variance (σ²). This is a super practical scenario because, in real-world situations, we often have sample data but lack complete knowledge about the entire population. In this article, we'll walk through how to construct a 99% confidence interval for μ given some sample data. We will use a sample of 144 observations, a sample mean (X̄) of 55.2, and a sample variance (S²) of 34.5. It sounds a bit complex, but trust me, we'll break it down step by step!

So, here’s the deal. We've got a random sample X1, X2, ..., X144 drawn from a distribution. The variance, Var[X], is unknown, but we denote it as σ². From our observed sample, we’ve calculated a sample mean (X̄) of 55.2 and a sample variance (S²) of 34.5. Our mission? Find a 99% confidence interval for μ, which represents the expected value E[Xi]. This type of problem pops up all the time in fields like market research, quality control, and even social sciences, where we need to estimate a population average based on a limited number of observations. Constructing this interval involves understanding the sampling distribution of the sample mean and using the t-distribution because the population variance is unknown. The t-distribution is crucial here because it accounts for the extra uncertainty introduced by estimating the population variance from the sample. We'll explore the degrees of freedom, which play a key role in determining the shape of the t-distribution and, consequently, the width of our confidence interval. Remember, a confidence interval gives us a range within which the true population mean is likely to fall, given our sample data and the level of confidence we choose (in this case, 99%).

Before we jump into the calculations, let’s quickly recap what a confidence interval actually means. A confidence interval gives us a range of values within which we believe the true population parameter (in this case, the mean μ) lies, with a certain level of confidence. Think of it as a net we cast out to catch the true mean. A 99% confidence interval means that if we were to repeat this sampling process 100 times, we would expect 99 of those intervals to contain the true population mean. It doesn't mean there's a 99% chance that the true mean is within this specific interval we calculate – rather, it's about the reliability of the method itself. The width of the interval is influenced by a few factors, such as the sample size, the sample variance, and the confidence level. A larger sample size generally leads to a narrower interval because we have more information about the population. A larger sample variance, on the other hand, tends to widen the interval because there's more variability in the data. And, of course, a higher confidence level (like 99% instead of 95%) results in a wider interval because we want to be more certain that we're capturing the true mean. So, in essence, when we calculate a 99% confidence interval, we're making a statement about our level of confidence in the procedure we're using to estimate the population mean, given the data we have at hand. It’s a cornerstone of statistical inference, allowing us to make informed decisions and draw meaningful conclusions from sample data.

Okay, let's get down to the nitty-gritty and calculate that 99% confidence interval. There are several steps involved, but don’t worry, we'll take it one at a time. First off, we need to identify the key pieces of information we have: the sample mean (X̄), the sample variance (S²), the sample size (n), and the desired confidence level. Next, we'll determine the degrees of freedom (df), which is simply n - 1. The degrees of freedom are important because they dictate the shape of the t-distribution we'll be using. After that, we'll find the critical t-value (tα/2) from the t-distribution table. This value corresponds to the level of confidence we want (99% in this case) and the degrees of freedom we calculated. The critical t-value is essentially a cutoff point that helps us define the boundaries of our confidence interval. Then, we'll calculate the standard error (SE), which measures the variability of the sample mean. It's calculated as the square root of the sample variance divided by the sample size. Finally, we'll plug all these values into the confidence interval formula: Confidence Interval = X̄ ± (tα/2 * SE). This formula gives us the lower and upper bounds of our interval, providing us with a range of plausible values for the population mean μ. It’s a bit like building a fence around our estimate, with the height of the fence determined by our confidence level and the variability in the data. So, let’s roll up our sleeves and get calculating! We'll go through each step meticulously to ensure we construct the confidence interval accurately.

Step-by-Step Calculation

Alright, let's break this down step-by-step so it's super clear. The first thing we need to do is gather all our known information. From the problem, we know the sample mean, often denoted as X̄, is 55.2. This is our best point estimate for the population mean. Next, we have the sample variance, S², which is 34.5. This tells us how spread out our sample data is. We also know the sample size, n, is 144, which is a pretty decent size and will help make our estimate more precise. Finally, we want a 99% confidence interval, which means our alpha (α) level is 1 - 0.99 = 0.01. This alpha level is crucial for finding the correct t-value. The next step is calculating the degrees of freedom (df). Remember, the formula for df is simply n - 1. So, in our case, df = 144 - 1 = 143. With the degrees of freedom in hand, we can now look up the critical t-value, often written as tα/2, in a t-distribution table. Since we have a two-tailed test (we’re looking for an interval, not just an upper or lower bound), we need to look up t0.005, 143. Now, a t-table typically doesn't have every single degree of freedom listed, especially for larger numbers like 143. We usually round down to the nearest available df, which might be 120 or even 100, depending on the table. For demonstration purposes, let's say t0.005, 143 is approximately 2.606 (this is a value you would get from a t-table or a calculator). This critical t-value tells us how many standard errors we need to extend from our sample mean to achieve our desired confidence level. Hang in there, we’re almost at the finish line!

Calculate Standard Error

Moving right along, the next key piece we need to calculate is the standard error (SE). The standard error is super important because it gives us an estimate of how much variability we can expect in our sample mean. In other words, it tells us how much our sample mean might bounce around if we took different samples from the same population. The formula for standard error when we don't know the population standard deviation (which is the case here) is SE = √(S²/n), where S² is the sample variance and n is the sample size. So, let's plug in the values we have. We know that S² is 34.5 and n is 144. Therefore, SE = √(34.5 / 144). When you calculate this, you get SE ≈ √(0.2396) ≈ 0.4895. This tells us that the standard error of the sample mean is approximately 0.4895. Think of this as the typical distance you'd expect a sample mean to be from the true population mean, just due to random chance. A smaller standard error means that our sample mean is likely to be closer to the true population mean, which is what we want! Now that we've got the standard error, we're in the home stretch. We have all the components we need to calculate the confidence interval. We have our sample mean, our critical t-value, and now our standard error. Next, we'll put all these pieces together to create the final interval. Keep following along, and you'll see how it all comes together!

Constructing the Confidence Interval

Okay, guys, we’re finally ready to put all the pieces together and build our 99% confidence interval! We've got the sample mean (X̄ = 55.2), the critical t-value (tα/2 ≈ 2.606), and the standard error (SE ≈ 0.4895). Now we just need to plug these values into the confidence interval formula, which is: Confidence Interval = X̄ ± (tα/2 * SE). This formula is the key to our answer. It tells us how to create a range around our sample mean that we're 99% confident contains the true population mean. Let’s break down what this formula means. The X̄ is our best guess for the population mean, based on our sample. The (tα/2 * SE) part is often called the margin of error. This margin of error accounts for the uncertainty in our estimate due to sampling variability. We multiply the critical t-value by the standard error to determine how wide our interval needs to be to achieve our desired confidence level. Now, let's plug in the numbers: Confidence Interval = 55.2 ± (2.606 * 0.4895). First, we calculate the margin of error: 2. 606 * 0.4895 ≈ 1.275. So, our margin of error is approximately 1.275. This means we're going to add and subtract 1.275 from our sample mean to get the upper and lower bounds of our interval. Next, we calculate the lower bound: 55.2 - 1.275 ≈ 53.925. And then, we calculate the upper bound: 55.2 + 1.275 ≈ 56.475. So, our 99% confidence interval for the population mean μ is approximately (53.925, 56.475). That wasn't so bad, right? We took it step by step, and now we have our result!

Woohoo! We did it! We've successfully calculated a 99% confidence interval for the population mean μ, and it's approximately (53.925, 56.475). So, what does this all mean? Well, we can be 99% confident that the true population mean falls within this range. Remember, this isn't a guarantee, but it's a statement about the reliability of our method. If we were to repeat this sampling process many times, 99% of the intervals we'd construct would contain the true population mean. This type of analysis is incredibly powerful because it allows us to make inferences about an entire population based on just a sample of data. We started with a random sample of 144 observations, calculated some statistics, and used the t-distribution to account for the unknown population variance. By following a clear, step-by-step approach, we were able to construct a confidence interval that gives us a meaningful estimate of the population mean. Confidence intervals are a fundamental tool in statistics, used across various fields to make informed decisions and draw conclusions from data. Whether you're in research, business, or any other data-driven field, understanding how to calculate and interpret confidence intervals is a crucial skill. So, the next time you encounter a similar problem, remember the steps we’ve covered, and you'll be well-equipped to tackle it! Keep up the great work, and stay curious about statistics!