Confidence Intervals: Finding Values Outside The Range

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Confidence Intervals: Finding Values Outside the Range

Hey guys! Let's dive into a cool math problem involving confidence intervals and see how we can figure out which values fall outside a specific range. We'll be working with a normally distributed population, which makes things a bit easier to handle. The core concept here is understanding how to build a confidence interval and then how to determine if a particular value is statistically plausible, given our data. This is super important stuff in statistics because it helps us make informed decisions and draw conclusions from samples. Ready to get started?

Understanding the Problem: Sample Mean, Standard Deviation, and Confidence Intervals

Okay, so the scenario we're tackling involves a simple random sample. We've got a sample size of 85 drawn from a normally distributed population. From this sample, we know the sample mean (146) and the sample standard deviation (34). Our mission? To determine which of the given values lies outside the 99% confidence interval for the population mean. Think of it like this: we're trying to create a range within which we're 99% confident the true population mean actually lives. Any value that falls outside of that range is, statistically speaking, pretty unusual, and that’s what we're looking for. The confidence interval itself is a range of values, calculated from sample data, which is likely to contain the true population parameter. The level of confidence, like our 99%, represents the probability that the interval contains the true population parameter. So, a 99% confidence interval means that if we were to take many samples and compute the confidence intervals for each sample, approximately 99% of those intervals would contain the true population mean. It's crucial for making accurate inferences about a population based on sample data. Understanding this helps us assess the reliability of our estimates and make well-informed decisions. Imagine you're a researcher studying something like the average height of a group of people. You can't measure every person, so you take a sample. The confidence interval gives you a range where you can be pretty sure the true average height for the entire group falls.

To construct this interval, we use the sample mean, the sample standard deviation, the sample size, and a critical value from the t-distribution or z-distribution (depending on whether the population standard deviation is known and the sample size). The formula for a confidence interval usually looks something like this: Sample Mean ± (Critical Value * (Standard Deviation / √(Sample Size)). That critical value is really important; it's what determines how wide the interval is and thus, our confidence level. A higher confidence level (like 99%) means a wider interval, as we need more room to be really sure we've captured the true population mean. Conversely, a lower confidence level (like 90%) gives a narrower interval. The sample size also plays a big role. A larger sample size generally leads to a narrower confidence interval, all other things being equal. This is because larger samples give us more information about the population, reducing the uncertainty in our estimate. So, what we're really doing is creating a safety net around our sample mean. It tells us how much our sample mean could reasonably vary due to chance. Now, let’s get down to the numbers, shall we?

Calculating the Confidence Interval

Alright, let’s get into the nitty-gritty of how to calculate this 99% confidence interval. Because we know the population is normally distributed, and our sample size is reasonably large (85), we can use the z-distribution. The z-distribution is a standard normal distribution, and we'll use a critical z-value that corresponds to our 99% confidence level. For a 99% confidence level, the z-score is approximately 2.576. This number tells us how many standard deviations away from the mean we need to go to capture 99% of the data. You can find this z-score using a standard z-table or a statistical calculator. Now, let’s calculate the standard error of the mean (SEM). The formula for the standard error of the mean is: Standard Deviation / √(Sample Size). In our case, that’s 34 / √85. Doing the math, √85 is roughly 9.22. So, 34 / 9.22 is approximately 3.69. The standard error is around 3.69. Now, we use the following formula to calculate the confidence interval:

  • Confidence Interval = Sample Mean ± (Z-score * Standard Error)

Plugging in our values: 146 ± (2.576 * 3.69) which is 146 ± 9.50. So, we add and subtract 9.50 from our sample mean to define the upper and lower bounds of our confidence interval. The lower bound of the 99% confidence interval is 146 - 9.50 = 136.50 and the upper bound is 146 + 9.50 = 155.50. This means we are 99% confident that the true population mean falls somewhere between 136.50 and 155.50. Any value outside of this range would be outside our confidence interval.

Identifying Values Outside the Interval

Okay, we’ve got our confidence interval: 136.50 to 155.50. The next step is to look at the list of provided values (which we don't have here, but let's assume some hypothetical values to illustrate). For the sake of demonstration, let’s imagine we have these values to assess: 130, 140, 150, 160. Remember, we are looking for the value that is outside our calculated range. Let's compare each hypothetical value:

  • 130: This value is less than 136.50 (our lower bound), so it falls outside the 99% confidence interval.
  • 140: This value is between 136.50 and 155.50, so it falls inside the confidence interval.
  • 150: This value is also within the confidence interval (between 136.50 and 155.50).
  • 160: This value is greater than 155.50 (our upper bound), so it also falls outside the confidence interval.

From these hypothetical values, both 130 and 160 are outside of the interval. If you were provided with actual options, you would simply compare them to your calculated confidence interval to find the values that don't fit within the bounds. This helps us to understand whether the values we're comparing are statistically plausible based on the information we have, or if they represent something unusual. Understanding this is key to making sound statistical inferences.

Significance of Values Outside the Confidence Interval

So, what does it mean when a value falls outside of our 99% confidence interval? Well, it suggests that the value is significantly different from what we'd expect based on our sample data. In other words, it's statistically unlikely that the true population mean is that value. If you were, for example, running an experiment and collected some additional data, and the new sample mean fell outside of your original confidence interval, you might have reason to question if something changed in your experiment, or if there’s some other factor at play that you hadn’t accounted for. This is where statistical significance comes into play. A value outside the 99% confidence interval is often considered to be statistically significant at the 1% significance level (because 100% - 99% = 1%). Statistical significance doesn’t necessarily mean the result is important in a practical sense, but it does mean that the results are unlikely to have happened by random chance alone. This becomes a cornerstone when we make conclusions about populations based on our sample data. The wider the confidence interval, the more room there is for the population mean to vary without being considered