Calculating 90% Confidence Intervals For Population Mean

In statistical analysis, determining the population mean is a crucial task, especially when dealing with normally distributed populations. However, obtaining data from the entire population is often impractical or impossible. Instead, we rely on samples drawn from the population to estimate the mean. This is where confidence intervals come into play. A confidence interval provides a range of values within which the true population mean is likely to lie, given a certain level of confidence. In this comprehensive guide, we will delve into the concept of a 90% confidence interval for the population mean when sampling from a normally distributed population, assuming the population standard deviation is unknown. We will also explore the crucial role of sample size, standard deviation, and the t-distribution in constructing these intervals. Specifically, we will consider a scenario where a simple random sample of size n is drawn from a normally distributed population, with the sample mean denoted as x̄ and the sample standard deviation as s. Our goal is to determine the 90% confidence interval for the true population mean, a fundamental task in statistical inference.

Core Concepts

Before diving into the calculation, let's solidify our understanding of the core concepts involved. The population mean (μ) is the average value of the variable of interest in the entire population. Since we cannot measure this directly, we estimate it using the sample mean (x̄), which is the average value in our sample. The sample standard deviation (s) measures the spread or variability of the data within the sample. The confidence level, in this case 90%, represents the probability that the true population mean falls within the calculated interval. This means that if we were to repeat the sampling process many times and construct 90% confidence intervals each time, approximately 90% of those intervals would contain the true population mean. The remaining 10% of the intervals would not capture the true mean, highlighting the inherent uncertainty in statistical estimation. This understanding of repeated sampling and the probabilistic interpretation of confidence intervals is critical for proper application and interpretation of statistical results. The degrees of freedom play a vital role in determining the appropriate t-value for confidence interval calculations. They reflect the amount of independent information available to estimate the population variance. In the context of a single sample t-test, the degrees of freedom are calculated as n - 1, where n is the sample size. This adjustment accounts for the fact that we are using the sample mean to estimate the population mean, thus losing one degree of freedom. The larger the degrees of freedom, the closer the t-distribution approximates the standard normal distribution.

Calculating the 90% Confidence Interval

To calculate the 90% confidence interval, we employ the t-distribution, a probability distribution similar to the standard normal distribution but with heavier tails. The t-distribution is particularly useful when the population standard deviation is unknown and we must rely on the sample standard deviation as an estimate. The formula for the confidence interval is:

Confidence Interval = x̄ ± (t-value * (s / √n))

Where:

• x̄ is the sample mean.
• s is the sample standard deviation.
• n is the sample size.
• t-value is the critical value from the t-distribution corresponding to the desired confidence level (90% in this case) and degrees of freedom (n-1).

The t-value is obtained from a t-distribution table or using statistical software. It depends on the desired confidence level and the degrees of freedom. For a 90% confidence level, we need to find the t-value that leaves 5% in each tail of the distribution (since 100% - 90% = 10%, and we divide this equally between the two tails). With n - 1 degrees of freedom, we look up the corresponding t-value in the t-table. This t-value essentially represents how many standard errors we need to extend from the sample mean to capture the true population mean with 90% confidence. A larger t-value indicates a wider confidence interval, reflecting greater uncertainty in our estimate.

Let's break down each component of the formula: The sample mean (x̄) serves as the point estimate for the population mean. It is the best single value we have to represent the true average of the population. However, it is unlikely to be exactly equal to the population mean due to sampling variability. The sample standard deviation (s) quantifies the spread of data within the sample. A larger standard deviation implies greater variability, which will lead to a wider confidence interval. The sample size (n) plays a crucial role in the precision of our estimate. A larger sample size generally results in a narrower confidence interval, as it provides more information about the population. The term s / √n is known as the standard error of the mean. It measures the variability of the sample means if we were to take multiple samples from the same population. The standard error decreases as the sample size increases, reflecting the improved precision of our estimate with larger samples. By multiplying the standard error by the t-value, we obtain the margin of error. This margin of error is the amount we add and subtract from the sample mean to create the confidence interval. It represents the maximum likely difference between the sample mean and the true population mean at the specified confidence level.

Step-by-Step Calculation

Let's illustrate the calculation with a concrete example. Suppose we have a sample of size n = 25, with a sample mean x̄ = 75 and a sample standard deviation s = 10. We want to construct a 90% confidence interval for the population mean.

1. Determine the degrees of freedom: Degrees of freedom = n - 1 = 25 - 1 = 24
2. Find the t-value: Using a t-table or statistical software, we find the t-value for a 90% confidence level and 24 degrees of freedom. This value is approximately 1.711.
3. Calculate the margin of error: Margin of error = t-value * (s / √n) = 1.711 * (10 / √25) = 1.711 * 2 = 3.422
4. Construct the confidence interval:
   • Lower limit = x̄ - margin of error = 75 - 3.422 = 71.578
   • Upper limit = x̄ + margin of error = 75 + 3.422 = 78.422

Therefore, the 90% confidence interval for the population mean is (71.578, 78.422). This means we are 90% confident that the true population mean lies within this range.

Interpreting the 90% Confidence Interval

The 90% confidence interval (71.578, 78.422) provides a range of plausible values for the population mean. It does not mean that there is a 90% probability that the true population mean is within this specific interval. Instead, it means that if we were to repeat the sampling process many times and construct 90% confidence intervals each time, approximately 90% of those intervals would contain the true population mean. The other 10% would not. The interpretation emphasizes the long-run performance of the confidence interval procedure rather than a probability statement about the specific interval calculated. It's crucial to avoid the common misinterpretation that the population mean has a 90% chance of being within the interval. The population mean is a fixed value, although unknown, and does not have a probability distribution. The uncertainty lies in our estimate based on the sample data. 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. Factors that influence the width of the interval include the sample size, the sample standard deviation, and the confidence level. A larger sample size will generally lead to a narrower interval, as will a smaller standard deviation. However, increasing the confidence level (e.g., from 90% to 95%) will result in a wider interval, as we need to capture a larger proportion of possible sample means.

Factors Affecting the Confidence Interval

Several factors can influence the width and accuracy of the confidence interval. Understanding these factors is crucial for designing effective studies and interpreting results appropriately.

• Sample Size (n): A larger sample size generally leads to a narrower confidence interval. This is because a larger sample provides more information about the population, reducing the uncertainty in our estimate. As n increases, the standard error of the mean (s / √n) decreases, which directly reduces the margin of error and the width of the interval.
• Sample Standard Deviation (s): A larger sample standard deviation indicates greater variability in the data, resulting in a wider confidence interval. When the data is more spread out, it is more difficult to pinpoint the true population mean, leading to a less precise estimate.
• Confidence Level: A higher confidence level (e.g., 95% instead of 90%) will result in a wider confidence interval. To be more confident that the interval captures the true population mean, we need to extend the range of plausible values. This is reflected in the larger t-value associated with higher confidence levels.
• Degrees of Freedom: The degrees of freedom (n - 1) influence the shape of the t-distribution. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution. With smaller degrees of freedom (smaller sample sizes), the t-distribution has heavier tails, resulting in larger t-values and wider confidence intervals.

Common Mistakes to Avoid

When working with confidence intervals, it's essential to avoid common pitfalls that can lead to misinterpretations and incorrect conclusions. Here are some frequent mistakes:

• Misinterpreting the Confidence Level: The most common mistake is interpreting the confidence level as the probability that the true population mean lies within the calculated interval. As previously discussed, the confidence level refers to the long-run proportion of intervals that would contain the true mean if the sampling process were repeated many times. It is not a probability statement about the specific interval calculated.
• Assuming Normality: The t-distribution method for constructing confidence intervals assumes that the population is normally distributed or that the sample size is large enough for the Central Limit Theorem to apply. If the population is severely non-normal and the sample size is small, the resulting confidence interval may not be accurate.
• Ignoring Outliers: Outliers can have a disproportionate impact on the sample mean and standard deviation, leading to a biased estimate of the population mean and a misleading confidence interval. It's important to identify and address outliers appropriately, either by removing them (if justified) or using robust statistical methods that are less sensitive to outliers.
• Confounding Statistical Significance with Practical Significance: A statistically significant result (i.e., a confidence interval that does not contain a specific value) does not necessarily imply practical significance. The magnitude of the effect and its real-world implications should also be considered. A very narrow confidence interval might be statistically significant but represent a trivial effect that is not meaningful in practice.

Conclusion

Calculating and interpreting a 90% confidence interval for the population mean is a fundamental skill in statistical inference. By understanding the underlying concepts, the calculation steps, and the factors that influence the interval, researchers and analysts can make informed decisions based on sample data. Remember to carefully consider the sample size, standard deviation, and confidence level when constructing and interpreting confidence intervals. By avoiding common mistakes and focusing on the practical implications of the results, you can effectively use confidence intervals to draw meaningful conclusions about the population of interest. This comprehensive understanding empowers you to move beyond point estimates and embrace the range of plausible values that confidence intervals provide, leading to more robust and reliable statistical analyses. In the realm of data-driven decision-making, the ability to construct and interpret confidence intervals is an invaluable asset. It allows us to quantify the uncertainty associated with our estimates and make informed judgments based on the available evidence. By mastering the concepts and techniques discussed in this guide, you can confidently navigate the complexities of statistical inference and extract meaningful insights from your data.

By grasping these essential steps and considering the factors that influence confidence intervals, you can effectively estimate population means and make informed decisions based on sample data.