Calculating The 90% Confidence Interval For Population Mean

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Introduction

In the realm of statistics, estimating population parameters from sample data is a common task. One crucial aspect of this estimation is determining the confidence interval, which provides a range of values within which the true population parameter is likely to lie. This article dives into how to calculate the confidence interval for a population mean when the population is normally distributed, and we have the sample mean, standard deviation, and sample size. So, if you've ever wondered how to put a range around your sample estimate, you're in the right place! Let's break it down in a way that's super easy to understand. Understanding confidence intervals is crucial because it allows us to make informed decisions based on sample data, acknowledging the inherent uncertainty involved. Instead of just providing a single point estimate, like the sample mean, a confidence interval gives us a range of plausible values for the population mean. This range helps us to better understand the potential variability in our estimate and to assess the precision of our findings. For instance, a narrower confidence interval suggests a more precise estimate, while a wider interval indicates greater uncertainty. Moreover, confidence intervals are used extensively in various fields such as healthcare, economics, and social sciences to draw meaningful conclusions and make reliable predictions. By grasping the methodology behind calculating confidence intervals, researchers and practitioners can effectively communicate the uncertainty associated with their estimates and make well-informed judgments.

Problem Statement

Imagine we've taken a simple random sample of 90 individuals from a population that we know follows a normal distribution. After crunching the numbers, we find that the average (mean) of our sample is 138, and the standard deviation, which tells us how spread out the data is, is 34. Now, we want to figure out a 90% confidence interval for the true average of the entire population. In simpler terms, we want to create a range of values within which we are 90% confident that the actual population mean lies. This range gives us a better understanding of the potential values for the population mean, rather than just a single point estimate. This is incredibly useful in various fields, from market research to medical studies, where understanding the possible range of values is more informative than a single estimate. We need to determine the lower and upper bounds of this interval, giving us a more complete picture of where the true population mean might be. This process involves using the sample statistics (mean and standard deviation), the sample size, and the appropriate critical value from the standard normal distribution (or t-distribution, depending on whether the population standard deviation is known). By calculating this confidence interval, we can make more informed decisions and draw more reliable conclusions based on our sample data.

Key Concepts

Before we dive into the calculations, let's clarify a few key concepts.

  • Population Mean: The true average value for the entire group we're interested in.
  • Sample Mean: The average value calculated from our sample, which is an estimate of the population mean.
  • Standard Deviation: A measure of how spread out the data is.
  • Confidence Interval: A range of values within which we are confident the population mean lies.
  • Confidence Level: The probability that the confidence interval contains the true population mean (e.g., 90%).
  • Critical Value: A value from a statistical distribution (like the z-distribution or t-distribution) that corresponds to our desired confidence level. This value helps us determine the margin of error.
  • Margin of Error: The amount added and subtracted from the sample mean to create the confidence interval. It reflects the uncertainty in our estimate due to sampling variability.

Understanding these concepts is crucial for anyone working with statistics, whether in academic research or practical applications. For instance, when conducting a survey, knowing the confidence interval helps us understand the range within which the true population opinion likely falls. In medical trials, confidence intervals around treatment effect estimates provide a more nuanced picture than just a point estimate, indicating the range of potential benefits or risks. Furthermore, in quality control, confidence intervals can help assess whether a production process is meeting certain standards. In each of these contexts, the ability to interpret and calculate confidence intervals is vital for making informed decisions and drawing valid conclusions.

Formula for Confidence Interval

The formula to calculate the confidence interval for a population mean (when the population standard deviation is unknown and we're using the sample standard deviation) is:

Confidence Interval = Sample Mean ± (Critical Value * (Sample Standard Deviation / √Sample Size))

Where:

  • Sample Mean is the average of our sample.
  • Critical Value is the z-score or t-score corresponding to our desired confidence level.
  • Sample Standard Deviation measures the spread of data in our sample.
  • Sample Size is the number of observations in our sample.

This formula essentially tells us how much to add and subtract from the sample mean to create a range that likely contains the population mean. The critical value, derived from either the z-distribution (for large sample sizes or known population standard deviation) or the t-distribution (for smaller sample sizes and unknown population standard deviation), determines the width of the interval. The standard error (sample standard deviation divided by the square root of the sample size) quantifies the uncertainty associated with our sample mean estimate. A larger sample size reduces the standard error, leading to a narrower and more precise confidence interval. This is because larger samples provide more information about the population, allowing us to estimate the population mean with greater accuracy. The margin of error, which is the product of the critical value and the standard error, represents the range around the sample mean within which we expect the population mean to lie with a certain level of confidence. By understanding and applying this formula, we can effectively estimate population parameters and communicate the uncertainty associated with our estimates.

Steps to Calculate the Confidence Interval

Let's apply this to our problem step-by-step:

Step 1: Identify the given values.

  • Sample Mean (xÌ„) = 138
  • Sample Standard Deviation (s) = 34
  • Sample Size (n) = 90
  • Confidence Level = 90% (which means α = 1 - 0.90 = 0.10)

Step 2: Determine the critical value.

Since our sample size is large (n = 90), we can use the z-distribution. For a 90% confidence level, we need to find the z-score that leaves 5% (0.05) in each tail of the distribution. Looking up the z-table (or using a calculator), we find that the z-score for 0.05 in the tail is approximately 1.645.

Step 3: Calculate the margin of error.

Margin of Error = Critical Value * (Sample Standard Deviation / √Sample Size) Margin of Error = 1.645 * (34 / √90) Margin of Error ≈ 1.645 * (34 / 9.487) Margin of Error ≈ 1.645 * 3.584 Margin of Error ≈ 5.896

Step 4: Calculate the confidence interval.

Confidence Interval = Sample Mean ± Margin of Error Confidence Interval = 138 ± 5.896 Lower Limit = 138 - 5.896 ≈ 132.104 Upper Limit = 138 + 5.896 ≈ 143.896

Therefore, the 90% confidence interval for the population mean is approximately (132.104, 143.896).

Breaking down the calculation into these steps makes it easier to understand and apply. First, identifying the given values sets the stage for the calculation by clarifying the known information. Determining the critical value is crucial because it dictates the width of the confidence interval, reflecting the level of confidence desired. The margin of error, calculated using the critical value, standard deviation, and sample size, represents the range of uncertainty around the sample mean. Finally, calculating the confidence interval by adding and subtracting the margin of error from the sample mean provides the range within which we expect the population mean to lie with the specified level of confidence. This step-by-step approach not only simplifies the calculation but also enhances understanding of the underlying principles and assumptions involved in constructing confidence intervals.

Interpretation

We can be 90% confident that the true population mean lies between 132.104 and 143.896. This means that if we were to take many samples and calculate confidence intervals for each, about 90% of those intervals would contain the true population mean. This range gives us a more realistic estimate than just the sample mean of 138, acknowledging the inherent uncertainty in using a sample to estimate a population parameter.

Interpreting confidence intervals correctly is essential for making sound statistical inferences. It's important to understand that the confidence level (90% in this case) refers to the method's long-run success rate, not the probability that the true population mean falls within this specific interval. In other words, if we were to repeat the sampling process many times and construct confidence intervals each time, approximately 90% of those intervals would capture the true population mean. The width of the confidence interval reflects the precision of our estimate; a narrower interval indicates a more precise estimate, while a wider interval suggests greater uncertainty. The interpretation also highlights the importance of considering the context of the data. For instance, a confidence interval might be used to estimate the average customer satisfaction score, the mean test score of students, or the average income in a particular region. Understanding what the data represents helps to make meaningful conclusions based on the confidence interval. By focusing on the practical implications and the uncertainty inherent in statistical estimation, we can use confidence intervals to make more informed decisions and draw more reliable conclusions.

Common Mistakes to Avoid

  • Misinterpreting the confidence level: It's not the probability that the true mean is in the interval, but rather the percentage of intervals that would contain the true mean if we repeated the process many times.
  • Using the wrong critical value: Make sure to use the appropriate distribution (z or t) and the correct degrees of freedom (if using the t-distribution).
  • Ignoring the assumptions: Confidence intervals assume a random sample and, for small samples, a normally distributed population.

Avoiding these common mistakes ensures the accurate calculation and interpretation of confidence intervals. One frequent error is misunderstanding the confidence level. The confidence level does not tell us the probability that the true population mean falls within the calculated interval; rather, it indicates the proportion of intervals that would contain the true mean if we repeated the sampling process multiple times. Another pitfall is selecting the wrong critical value. It's crucial to distinguish between when to use the z-distribution (typically for large sample sizes or when the population standard deviation is known) and the t-distribution (for smaller sample sizes when the population standard deviation is unknown). Additionally, when using the t-distribution, it is essential to consider the correct degrees of freedom, which affect the shape of the distribution and, consequently, the critical value. Overlooking the assumptions underlying the use of confidence intervals is another common mistake. These assumptions often include that the sample was randomly selected from the population and, for small sample sizes, that the population is approximately normally distributed. Violating these assumptions can lead to inaccurate confidence intervals and misleading inferences. By being aware of these potential errors and taking steps to avoid them, we can ensure that our confidence intervals are both valid and meaningful.

Conclusion

Calculating the confidence interval for a population mean is a fundamental statistical technique. By following these steps, we can estimate the range within which the true population mean likely lies. In our example, we are 90% confident that the population mean is between 132.104 and 143.896. Remember, confidence intervals are a powerful tool for making informed decisions based on sample data.

In conclusion, mastering the calculation of confidence intervals for population means is an invaluable skill in statistics. By systematically working through the steps, we can derive a meaningful range that likely captures the true population mean. In our illustrative example, we successfully calculated a 90% confidence interval, providing us with a range of 132.104 to 143.896 within which we are 90% confident the population mean resides. This range offers a more informative perspective than a single point estimate, acknowledging the uncertainty inherent in sampling. Confidence intervals empower us to make data-driven decisions, guiding us to understand the potential variability and precision of our estimates. The ability to compute and interpret confidence intervals allows researchers and practitioners to communicate their findings more effectively, highlighting the range of plausible values for population parameters. Whether in academic research, business analysis, or public health, confidence intervals are indispensable for drawing sound conclusions and making well-informed judgments based on sample data. The use of confidence intervals enhances the robustness and reliability of statistical inferences, making them a cornerstone of evidence-based decision-making.