Calculating 98% Confidence Interval For Population Mean

In statistical analysis, a primary goal is often to estimate population parameters based on sample data. The population mean, a crucial measure of central tendency, is frequently the target of such estimations. However, since it's impractical to survey an entire population, we rely on samples to infer characteristics about the whole group. This is where the concept of a confidence interval becomes indispensable. A confidence interval provides a range within which the true population mean is likely to fall, given a certain level of confidence. This article will guide you through the process of calculating a 98% confidence interval for a population mean, using a sample size of 51, a sample mean of 38.9, and a standard deviation of 6.5. Understanding this calculation is crucial for anyone working with data analysis, as it allows for more informed and reliable conclusions about population characteristics.

Understanding Confidence Intervals

Before diving into the calculations, it's essential to grasp the fundamental concept of confidence intervals. A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. The confidence level, expressed as a percentage (e.g., 90%, 95%, 98%, 99%), indicates the probability that the interval captures the true population parameter if we were to repeat the sampling process multiple times. For example, a 95% confidence interval suggests that if we were to draw 100 different samples and construct confidence intervals for each, approximately 95 of those intervals would contain the true population mean. The higher the confidence level, the wider the interval, reflecting a greater degree of certainty that the true parameter is included. However, a wider interval also implies less precision in our estimate. Conversely, a narrower interval provides a more precise estimate but comes with a lower confidence level. The balance between precision and confidence is a critical consideration when interpreting and applying confidence intervals.

Key Components for Confidence Interval Calculation

To calculate a confidence interval for a population mean, several key components are required. First, the sample mean is the average value calculated from the sample data and serves as the point estimate for the population mean. In our case, the sample mean is 38.9. Second, the sample standard deviation measures the dispersion or variability of the data points within the sample. A larger standard deviation indicates greater variability. In this scenario, the sample standard deviation is 6.5. Third, the sample size refers to the number of observations in the sample, which is 51 in our example. The sample size plays a crucial role in determining the margin of error and the width of the confidence interval. Larger sample sizes generally lead to narrower intervals, providing more precise estimates. Finally, the confidence level determines the critical value used in the calculation. For a 98% confidence level, we need to find the appropriate critical value (z-score or t-score) that corresponds to the desired level of confidence.

Determining the Appropriate Critical Value

The critical value is a crucial factor in calculating the confidence interval. It is determined by the confidence level and the distribution of the sample data. When the population standard deviation is unknown, and the sample size is relatively small (typically less than 30), we use the t-distribution. However, with a sample size of 51, which is considered large enough, we can use the standard normal distribution (z-distribution) as an approximation, even if the population standard deviation is unknown. The critical value corresponds to the number of standard deviations away from the mean that encompass the desired confidence level. For a 98% confidence level, we need to find the z-score that leaves 1% in each tail of the distribution (since 100% - 98% = 2%, and 2% / 2 = 1%). Using a z-table or a statistical calculator, we find that the z-score corresponding to a 98% confidence level is approximately 2.326. This value signifies that we are 98% confident that the true population mean lies within 2.326 standard errors of the sample mean.

Calculating the Margin of Error

The margin of error is a critical component in determining the width of the confidence interval. It represents the amount by which the sample mean might differ from the true population mean. The margin of error is calculated by multiplying the critical value by the standard error of the mean. The standard error of the mean, in turn, is calculated by dividing the sample standard deviation by the square root of the sample size. In our case, the sample standard deviation is 6.5, and the sample size is 51. Therefore, the standard error of the mean is 6.5 / √51 ≈ 0.908. Now, we multiply this standard error by the critical value (2.326) to obtain the margin of error: 2.326 * 0.908 ≈ 2.112. This margin of error indicates that we can expect the sample mean to be within approximately 2.112 units of the true population mean with 98% confidence. A smaller margin of error results in a narrower confidence interval, providing a more precise estimate of the population mean.

Constructing the 98% Confidence Interval

Now that we have calculated the sample mean (38.9), the critical value (2.326), and the margin of error (2.112), we can construct the 98% confidence interval. The confidence interval is calculated by adding and subtracting the margin of error from the sample mean. The lower limit of the interval is the sample mean minus the margin of error: 38.9 - 2.112 ≈ 36.788. The upper limit of the interval is the sample mean plus the margin of error: 38.9 + 2.112 ≈ 41.012. Therefore, the 98% confidence interval for the population mean is approximately (36.788, 41.012). This means we are 98% confident that the true population mean lies within this range. It's essential to interpret this interval correctly. It does not mean that there is a 98% chance that the true population mean is within this specific interval. Rather, it means that if we were to take many samples and construct confidence intervals in the same way, 98% of those intervals would contain the true population mean.

Interpreting the Results and Conclusion

The 98% confidence interval, calculated as (36.788, 41.012), provides a range within which we can be highly confident that the true population mean lies. This interval suggests that the true average value for the entire population is likely to be between 36.788 and 41.012, based on the sample data. The interpretation of this confidence interval is crucial for making informed decisions and drawing meaningful conclusions from the data. It's important to remember that this interval is an estimate, and there is a 2% chance that the true population mean falls outside this range. However, the high confidence level (98%) indicates a strong likelihood that the interval captures the true parameter. In conclusion, by understanding and applying the principles of confidence interval calculation, we can effectively estimate population parameters from sample data, providing valuable insights for various research and practical applications. The calculated 98% confidence interval (36.788, 41.012) gives us a reliable range within which the true population mean is likely to reside, based on the given sample statistics.