80% Confidence Interval: House Square Footage

by Admin 46 views
80% Confidence Interval: House Square Footage

Hey guys! Let's dive into calculating a confidence interval for the average square footage of houses. This is super useful when you want to estimate a range where the true average likely falls, based on a sample of houses. We're going to use some statistics, but don't worry, I'll break it down simply. So, grab your calculators, and let’s get started!

Understanding Confidence Intervals

First off, what exactly is a confidence interval? Think of it as a range of values that you're pretty sure contains the true population mean. In our case, it’s the average square footage of all houses. We're aiming for an 80% confidence interval, which means we want to be 80% confident that the true average square footage lies within our calculated range. The higher the confidence level, the wider the interval, because you want to be more certain you're capturing the true mean. Now, why do we need confidence intervals? Well, unless we measure every single house (which is usually impossible), we can only rely on a sample. A sample gives us an estimate, but there’s always a margin of error. The confidence interval helps us quantify that margin of error and gives us a more realistic picture of the true population mean. Remember, the population mean is the average square footage of all houses, while the sample mean is the average of the houses we actually measured. The confidence interval bridges the gap between these two.

Gathering the Data

Alright, let's gather the information we need for our calculation. The problem gives us the following key pieces of data, and it's crucial to understand each one:

  • Population Standard Deviation (σ): 154 square feet. This tells us how spread out the square footage values are in the entire population of houses. A larger standard deviation means more variability.
  • Sample Size (n): 16 houses. This is the number of houses we included in our sample. The larger the sample size, the more accurate our estimate of the population mean tends to be.
  • Sample Mean (xÌ„): 1550 square feet. This is the average square footage of the 16 houses in our sample. It's our best point estimate for the true average square footage.
  • Confidence Level: 80%. This tells us how confident we want to be that our interval contains the true population mean. In other words, if we were to repeat this sampling process many times, 80% of the confidence intervals we construct would contain the true population mean.

Before we move on, let's quickly recap what each of these terms means. The population standard deviation measures the spread of data in the entire population. The sample size is the number of observations in our sample. The sample mean is the average of our sample, and the confidence level reflects our certainty that the true population mean lies within our interval. Having a clear understanding of these terms is essential for accurately calculating and interpreting our confidence interval. Now that we have all our data, we can move on to the next step: finding the critical value.

Finding the Critical Value (Z-score)

Okay, so we need to find the critical value, often called the Z-score. This value is derived from the standard normal distribution and corresponds to our desired confidence level of 80%. In simpler terms, the Z-score tells us how many standard deviations away from the mean we need to go to capture 80% of the area under the normal distribution curve. For an 80% confidence level, we need to find the Z-score that leaves 10% in each tail of the distribution (since 100% - 80% = 20%, and we split that evenly between the two tails). You can find this Z-score using a Z-table or a calculator with statistical functions. A Z-table typically shows the area to the left of a given Z-score. Since we want the area to the left of the Z-score to be 0.90 (to leave 0.10 in the right tail), we look for 0.90 in the Z-table. The closest value we find is usually around 1.28. So, our Z-score (critical value) is approximately 1.28. This means that to achieve an 80% confidence level, we need to extend our interval about 1.28 standard deviations from the sample mean. Keep in mind that the Z-score represents the number of standard deviations from the mean in a standard normal distribution. It's a crucial value because it helps us determine the margin of error for our confidence interval. With the Z-score in hand, we can now calculate the margin of error and construct our confidence interval.

Calculating the Margin of Error

Alright, now that we've got our Z-score, let's calculate the margin of error. The margin of error tells us how much we need to add and subtract from our sample mean to create our confidence interval. The formula for the margin of error (E) is:

E = Z * (σ / √n)

Where:

  • Z is the Z-score (critical value) = 1.28
  • σ is the population standard deviation = 154
  • n is the sample size = 16

Plugging in the values, we get:

E = 1.28 * (154 / √16) E = 1.28 * (154 / 4) E = 1.28 * 38.5 E ≈ 49.28

So, our margin of error is approximately 49.28 square feet. This means that we'll add and subtract 49.28 from our sample mean to get the upper and lower bounds of our confidence interval. The margin of error is directly influenced by the Z-score, the population standard deviation, and the sample size. A larger Z-score (higher confidence level) or a larger population standard deviation will result in a larger margin of error. Conversely, a larger sample size will result in a smaller margin of error. This makes sense because a larger sample size provides more information and reduces the uncertainty in our estimate of the population mean. Now that we have the margin of error, we can finally construct our confidence interval.

Constructing the Confidence Interval

Okay, we're in the home stretch! Now we use the margin of error to build the confidence interval. Remember, the confidence interval is a range within which we believe the true population mean lies with a certain level of confidence (in this case, 80%). The formula for the confidence interval is:

Confidence Interval = (Sample Mean - Margin of Error, Sample Mean + Margin of Error)

We know:

  • Sample Mean (xÌ„) = 1550
  • Margin of Error (E) ≈ 49.28

So, let's plug those values in:

Lower Bound = 1550 - 49.28 ≈ 1500.72 Upper Bound = 1550 + 49.28 ≈ 1599.28

Therefore, our 80% confidence interval for the population mean of house square footage is approximately (1500.72, 1599.28) square feet. This means we are 80% confident that the true average square footage of all houses falls between 1500.72 and 1599.28 square feet. To summarize, we started with a sample of 16 houses and found their average square footage to be 1550 square feet. Using the population standard deviation and our desired confidence level, we calculated a margin of error and constructed a confidence interval. This interval gives us a range within which we can reasonably expect the true population mean to lie. Remember, the confidence interval is not a guarantee, but it provides a valuable estimate based on the available data. Now, let's interpret these results in a meaningful way.

Interpreting the Results

Alright, so we've crunched the numbers and found that our 80% confidence interval for the average square footage of houses is (1500.72, 1599.28) square feet. But what does this actually mean? Well, it means we're 80% confident that the true average square footage of all houses in the population falls somewhere between 1500.72 and 1599.28 square feet. Think of it this way: if we were to take many different samples of 16 houses and calculate a confidence interval for each sample, about 80% of those intervals would contain the true population mean. It's important to remember that this doesn't mean there's an 80% chance that the true mean falls within this specific interval. The true mean is a fixed value, and it either is or isn't within our interval. Rather, the 80% confidence level refers to the reliability of our method. Our method of constructing confidence intervals will capture the true mean 80% of the time. So, while we can't say for sure that the true average square footage is within our interval, we can be reasonably confident that it is. This information can be incredibly useful for various purposes. For example, real estate agents could use this information to provide potential buyers with a realistic range of expected house sizes. Developers could use it to plan the construction of new homes that meet the needs of the population. And homeowners could use it to compare the size of their homes to the average in their area.

Conclusion

So there you have it! We've successfully calculated and interpreted an 80% confidence interval for the average square footage of houses. We started with a sample of 16 houses, found the sample mean, and used the population standard deviation and Z-score to calculate the margin of error. Finally, we constructed the confidence interval, which gave us a range within which we can be reasonably confident that the true population mean lies. Remember, confidence intervals are a powerful tool for making inferences about populations based on sample data. They allow us to quantify the uncertainty in our estimates and provide a more realistic picture of the true population parameters. By understanding how to calculate and interpret confidence intervals, you can make more informed decisions and draw more meaningful conclusions from data. Whether you're a student learning statistics, a researcher analyzing data, or a professional making business decisions, confidence intervals can be a valuable asset in your toolkit. Keep practicing, and you'll become a pro at calculating and interpreting them in no time! Great job, guys! You nailed it!