Margin Of Error Calculation For Airline Travel Time With 90% Confidence

In this article, guys, we're diving into a common statistical problem: calculating the margin of error. Specifically, we'll be looking at a scenario involving airline travelers. Imagine we've surveyed 320 randomly selected travelers and found that, on average, they spend 24 hours traveling per year, with a standard deviation of 2.9 hours. Our goal is to determine the margin of error, assuming a 90% confidence level. This is super important because it helps us understand the range within which the true population mean likely falls. So, let's break it down step by step, making sure we not only get the answer but also understand the why behind the calculations. Grab your thinking caps, and let's get started!

Okay, so what exactly is the margin of error? In simple terms, the margin of error is the range above and below the sample statistic (like our sample mean of 24 hours) within which the true population parameter (the true average travel time for all airline travelers) is likely to fall. It's like a buffer zone that accounts for the uncertainty that comes with using a sample to estimate something about a larger population. Think of it this way: we can't survey every single airline traveler in the world, so we take a sample. But because it's just a sample, our results might not perfectly reflect the entire population. The margin of error gives us a sense of how much our sample result might differ from the true population value. The size of the margin of error depends on a few things, including the sample size, the variability in the sample (measured by the standard deviation), and the confidence level we want to have. A higher confidence level means we want to be more certain that the true population value falls within our calculated range, which, in turn, leads to a larger margin of error. This is because we're casting a wider net to capture the true value. Conversely, a larger sample size generally leads to a smaller margin of error because a bigger sample gives us a more precise estimate of the population. So, in our case, we want to find this buffer zone for the average travel time, knowing that we've surveyed 320 travelers and want to be 90% confident in our estimate. The margin of error essentially tells us how much wiggle room we have around our sample mean to make a confident statement about the population mean. Without understanding the margin of error, it's tough to make meaningful inferences from sample data because we'd have no idea how reliable our estimates are. It's a crucial concept in statistics and helps us make informed decisions based on data.

Before we jump into the calculation, let's identify the key components we need. We have the sample mean (${\bar{x}}$), which is 24 hours. This is the average travel time we found from our sample of 320 travelers. We also have the sample standard deviation (s), which is 2.9 hours. This tells us how much the individual travel times in our sample vary around the mean. A larger standard deviation means the data points are more spread out, while a smaller standard deviation means they are clustered closer to the mean. The sample size (n) is 320, which is the number of travelers we surveyed. A larger sample size generally leads to a more precise estimate and a smaller margin of error. Finally, we have the confidence level, which is 90%. This means we want to be 90% confident that the true population mean falls within our calculated interval. The confidence level is closely related to the critical value (\z^*), which we'll need to find. The critical value is a Z-score that corresponds to the desired confidence level. For a 90% confidence level, the critical value is approximately 1.645. This value comes from the standard normal distribution and represents the number of standard deviations away from the mean that we need to go to capture 90% of the data. To recap, we have:\

• Sample Mean (\bar{x}): 24 hours
• Standard Deviation (s): 2.9 hours
• Sample Size (n): 320
• Confidence Level: 90%
• Critical Value (\z^*): 1.645

With these components in hand, we're ready to plug them into the formula for the margin of error. Understanding these components is crucial because they each play a role in determining the size of the margin of error. The sample mean gives us a central point, the standard deviation tells us about the variability, the sample size influences the precision, and the confidence level dictates how certain we want to be. By carefully considering each of these, we can calculate a margin of error that accurately reflects the uncertainty in our estimate.

Alright, now that we've got all the pieces, let's put them together using the formula for the margin of error. The formula we'll use here is specifically for estimating the population mean when we know the sample standard deviation. It looks like this: ${\text{Margin of Error} = z^* \times \frac{s}{\sqrt{n}}}$ Where:\

• (\z^*) is the critical value corresponding to our desired confidence level.
• (s) is the sample standard deviation.
• (n) is the sample size.

Let's break down what this formula is telling us. The critical value (\z^*) is like a multiplier that adjusts the margin of error based on how confident we want to be. A higher confidence level means a larger critical value, which results in a larger margin of error. This makes sense because if we want to be more confident, we need to cast a wider net. The sample standard deviation (s) reflects the variability in our data. If the data points are more spread out (higher standard deviation), the margin of error will be larger because there's more uncertainty in our estimate. Conversely, if the data points are clustered closely together (lower standard deviation), the margin of error will be smaller. The square root of the sample size (\sqrt{n}) is in the denominator, which means that as the sample size increases, the margin of error decreases. This is because a larger sample gives us a more precise estimate of the population, reducing the uncertainty. So, to calculate the margin of error, we simply multiply the critical value by the standard error (which is ${\frac{s}{\sqrt{n}}}$). This gives us the amount we need to add and subtract from the sample mean to create our confidence interval. This formula is a powerful tool for quantifying the uncertainty in our estimates and making informed decisions based on data. By understanding the role of each component, we can better interpret the margin of error and its implications for our analysis.

Okay, let's get down to the nitty-gritty and calculate that margin of error! We've already identified all the components we need, so now it's just a matter of plugging them into the formula. Remember, the formula is: ${\text{Margin of Error} = z^* \times \frac{s}{\sqrt{n}}}$ Here are the steps we'll follow:

1. Identify the values:
   • Critical Value (\z^*): 1.645
   • Standard Deviation (s): 2.9 hours
   • Sample Size (n): 320
2. Plug the values into the formula: ${\text{Margin of Error} = 1.645 \times \frac{2.9}{\sqrt{320}}}$
3. Calculate the square root of the sample size: ${\sqrt{320} \approx 17.8885}$
4. Divide the standard deviation by the square root of the sample size: ${\frac{2.9}{17.8885} \approx 0.1621}$
5. Multiply the critical value by the result from step 4: ${1.645 \times 0.1621 \approx 0.2666}$
6. Round the answer to the nearest tenth: ${\text{Margin of Error} \approx 0.3}$

So, the margin of error is approximately 0.3 hours. This means that we can be 90% confident that the true average travel time for all airline travelers falls within 0.3 hours of our sample mean of 24 hours. Breaking the calculation down into these steps makes it easier to follow and understand. We started by identifying all the necessary values, then systematically plugged them into the formula, performed the calculations, and rounded our final answer. This step-by-step approach ensures accuracy and helps us avoid errors. Now that we have our margin of error, we can use it to construct a confidence interval, which will give us a range within which the true population mean is likely to lie. But we will keep that for the next section.

Now that we've calculated the margin of error to be approximately 0.3 hours, let's talk about what this actually means. In simple terms, it tells us the range of uncertainty around our sample mean. We found that the average travel time for our sample of 320 airline travelers was 24 hours. The margin of error of 0.3 hours means that we can be 90% confident that the true average travel time for all airline travelers falls within the range of 24 hours plus or minus 0.3 hours. This gives us a confidence interval of (23.7 hours, 24.3 hours). The confidence interval is a range of values that we believe, with a certain level of confidence, contains the true population parameter. In this case, we're 90% confident that the true average travel time for all airline travelers is somewhere between 23.7 and 24.3 hours. It's super important to understand that this doesn't mean there's a 90% chance that the true mean is within this interval. Instead, it means that if we were to repeat this survey many times, 90% of the confidence intervals we construct would contain the true population mean. The other 10% of the intervals would not capture the true mean, which is why we can't be 100% certain. The margin of error is crucial for making informed decisions based on data. Without it, we'd only have our sample mean, which is just a single point estimate and doesn't tell us anything about the uncertainty involved. By including the margin of error, we acknowledge that our sample might not perfectly represent the entire population and provide a range of plausible values for the true population mean. So, in the context of our airline travel example, the margin of error helps us understand that while our sample suggests an average travel time of 24 hours, the true average could reasonably be a bit lower or higher. This information is valuable for airlines, travel agencies, and anyone else interested in understanding travel patterns. The interpretation of the margin of error is just as important as the calculation itself. It's what allows us to translate a statistical result into meaningful insights and make informed decisions.

Alright, guys, we've made it to the end! We've successfully calculated the margin of error for the average travel time of airline travelers, and more importantly, we've understood why we did it and what it means. To recap, we started with a sample of 320 travelers, found a sample mean of 24 hours, and a standard deviation of 2.9 hours. We wanted to find the margin of error at a 90% confidence level. We identified all the necessary components: the critical value (1.645), the standard deviation (2.9 hours), and the sample size (320). We then plugged these values into the formula for the margin of error: ${\text{Margin of Error} = z^* \times \frac{s}{\sqrt{n}}}$. After performing the calculations, we found the margin of error to be approximately 0.3 hours. This means we can be 90% confident that the true average travel time for all airline travelers falls within the range of 23.7 to 24.3 hours. Understanding the margin of error is crucial for making informed decisions based on statistical data. It helps us quantify the uncertainty in our estimates and provides a range of plausible values for the true population parameter. Without it, we'd only have a single point estimate, which doesn't tell us anything about the variability or reliability of our result. So, the next time you're looking at survey data or research findings, remember the margin of error! It's a key piece of information that helps you interpret the results accurately and make sound judgments. And remember, statistics isn't just about numbers; it's about understanding the story they tell. By understanding concepts like the margin of error, we can become more critical and informed consumers of data. Great job, everyone, for sticking with it! You've now added another valuable tool to your statistical toolkit.