Determine If 48.8 Is Within The 90% Confidence Interval Given A Sample Size Of 225, A Mean Of 48.5, And A Standard Deviation Of 1.8.

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In statistical analysis, confidence intervals provide a range of values within which the true population parameter is likely to lie. When analyzing sample data, it's crucial to understand how to construct and interpret these intervals. In this article, we will delve into the process of determining whether a specific value, in this case, 48.8, falls within a 90% confidence interval calculated from a sample with a mean of 48.5, a standard deviation of 1.8, and a sample size of 225. We'll break down each step involved, from calculating the margin of error to interpreting the final interval. This comprehensive guide will equip you with the knowledge to confidently assess whether a given value is plausible within the context of your sample data. We will explore the significance of confidence levels, the impact of sample size and standard deviation, and the practical implications of these calculations. Furthermore, we will emphasize the importance of understanding the underlying assumptions and limitations of confidence intervals, ensuring a robust and accurate interpretation of your results. This detailed explanation is designed to provide clarity and practical application for both students and professionals alike.

Understanding Confidence Intervals

To determine if 48.8 is within the 90% confidence interval, we first need to understand what a confidence interval represents. A confidence interval is a range of values that, with a certain level of confidence, contains the true population mean. The 90% confidence level indicates that if we were to take many samples and construct confidence intervals from each, approximately 90% of those intervals would contain the true population mean. The confidence interval is a fundamental concept in inferential statistics, allowing us to make informed estimations about population parameters based on sample data. The construction of a confidence interval takes into account several key factors, including the sample mean, the sample standard deviation, the sample size, and the desired level of confidence. Each of these elements plays a crucial role in determining the width and position of the interval. A larger sample size generally leads to a narrower interval, reflecting greater precision in our estimate. Conversely, a higher standard deviation results in a wider interval, indicating greater variability in the data. The confidence level directly impacts the width of the interval, with higher confidence levels (e.g., 95% or 99%) producing wider intervals than lower confidence levels (e.g., 90%). Understanding these relationships is essential for interpreting and applying confidence intervals effectively. In practical terms, a confidence interval provides a range of plausible values for the population mean, offering a more comprehensive picture than a single point estimate. This range is particularly valuable in decision-making processes, as it acknowledges the inherent uncertainty in statistical inference. By considering the width and position of the confidence interval, we can assess the strength of the evidence and the potential range of true population values. The concept of confidence intervals is widely applied across various fields, including healthcare, finance, engineering, and social sciences, making it a cornerstone of statistical analysis.

Calculating the Margin of Error

The margin of error is a critical component in constructing a confidence interval. It quantifies the amount of uncertainty associated with our estimate of the population mean. The margin of error is calculated by multiplying the critical value (z-score or t-score) by the standard error. In this case, since we have a large sample size (n = 225) and the population standard deviation is unknown, we can use the z-score. First, we need to find the appropriate z-score for a 90% confidence level. For a 90% confidence level, the alpha level (α) is 1 - 0.90 = 0.10. Since we are constructing a two-tailed confidence interval, we divide alpha by 2, resulting in α/2 = 0.05. The z-score that corresponds to a cumulative probability of 1 - 0.05 = 0.95 is approximately 1.645. The z-score is a standardized value that represents the number of standard deviations a data point is from the mean in a standard normal distribution. The choice of the z-score or t-score depends on the sample size and whether the population standard deviation is known. For large sample sizes (typically n > 30), the z-score is a reasonable approximation, even if the population standard deviation is unknown, because the sample standard deviation provides a reliable estimate. The standard error, which measures the variability of the sample mean, is calculated by dividing the sample standard deviation by the square root of the sample size. In this scenario, the standard error is 1.8 / √225 = 1.8 / 15 = 0.12. Now, we can calculate the margin of error by multiplying the z-score (1.645) by the standard error (0.12), which yields a margin of error of approximately 0.1974. The margin of error represents the maximum likely difference between the sample mean and the true population mean at the specified confidence level. A smaller margin of error indicates a more precise estimate, while a larger margin of error reflects greater uncertainty. Understanding how to calculate and interpret the margin of error is crucial for constructing and evaluating confidence intervals effectively.

Determining the Confidence Interval

With the margin of error calculated, we can now construct the 90% confidence interval. The confidence interval is determined by adding and subtracting the margin of error from the sample mean. The sample mean is given as 48.5, and the calculated margin of error is approximately 0.1974. To find the lower bound of the confidence interval, we subtract the margin of error from the sample mean: 48.5 - 0.1974 = 48.3026. For the upper bound, we add the margin of error to the sample mean: 48.5 + 0.1974 = 48.6974. Therefore, the 90% confidence interval is approximately (48.3026, 48.6974). This means that we are 90% confident that the true population mean falls within this range. The confidence interval provides a range of plausible values for the population mean, reflecting the uncertainty inherent in statistical inference. The width of the interval is directly influenced by the margin of error, which in turn is affected by the sample size, standard deviation, and confidence level. A narrower confidence interval indicates a more precise estimate of the population mean, while a wider interval suggests greater variability or uncertainty. Interpreting the confidence interval correctly is essential for making informed decisions based on sample data. The interval should not be interpreted as the probability that the true population mean falls within the calculated range; rather, it represents the proportion of intervals constructed from repeated samples that would contain the true mean. In this case, if we were to take many samples and construct 90% confidence intervals for each, we would expect approximately 90% of those intervals to include the true population mean. The confidence interval is a powerful tool for statistical inference, providing a range of plausible values for the population parameter and allowing us to quantify the uncertainty associated with our estimate.

Is 48.8 Within the Confidence Interval?

Now that we have calculated the 90% confidence interval (48.3026, 48.6974), we can determine whether 48.8 falls within this range. By examining the interval, we can see that 48.8 is greater than the upper bound of the interval, which is 48.6974. Therefore, 48.8 is not within the 90% confidence interval. This observation has important implications for our understanding of the data. If a value falls outside the confidence interval, it suggests that it is unlikely to be the true population mean, given the sample data. In other words, the sample data provides evidence against the hypothesis that the population mean is 48.8. The confidence interval serves as a critical tool for hypothesis testing, allowing us to assess the plausibility of different values for the population parameter. Values within the interval are considered plausible, while those outside the interval are considered less likely. The decision to reject a particular value as the population mean is based on the chosen confidence level. A higher confidence level results in a wider confidence interval, making it more likely that a given value will fall within the interval. Conversely, a lower confidence level produces a narrower confidence interval, increasing the likelihood that a value will fall outside the interval. In this scenario, the fact that 48.8 lies outside the 90% confidence interval indicates that it is a relatively extreme value compared to what we would expect if the true population mean were within the calculated range. This finding might prompt further investigation, such as examining the sample data for potential biases or considering alternative hypotheses about the population mean. The confidence interval provides a clear and intuitive framework for assessing the compatibility of sample data with different possible values of the population parameter.

Implications and Interpretation

The fact that 48.8 is not within the 90% confidence interval (48.3026, 48.6974) has significant implications. It suggests that the true population mean is likely lower than 48.8. This conclusion is based on the sample data and the chosen confidence level. The confidence interval provides a range of plausible values for the population mean, and any value outside this range is considered less likely. When interpreting confidence intervals, it's important to remember that they do not provide a definitive statement about the true population mean. Instead, they offer a probabilistic range within which the mean is likely to fall, given the sample data. The 90% confidence level indicates that if we were to repeat the sampling process many times and construct confidence intervals each time, approximately 90% of those intervals would contain the true population mean. This does not mean that there is a 90% probability that the true mean falls within the specific interval we calculated. The true population mean is a fixed value, and the confidence interval is a range that varies from sample to sample. The interpretation of a confidence interval also depends on the context of the study. In practical applications, the confidence interval can inform decision-making by providing a range of plausible values for the parameter of interest. If a specific value is of particular importance, we can assess whether it falls within the confidence interval to determine its compatibility with the sample data. The width of the confidence interval reflects the precision of our estimate. A narrow confidence interval suggests that we have a more precise estimate of the population mean, while a wide confidence interval indicates greater uncertainty. The sample size, standard deviation, and confidence level all influence the width of the confidence interval. Understanding these factors is crucial for interpreting confidence intervals effectively. In summary, the confidence interval is a valuable tool for statistical inference, providing a range of plausible values for the population parameter and allowing us to quantify the uncertainty associated with our estimate. The fact that 48.8 is not within the calculated 90% confidence interval suggests that it is an unlikely value for the true population mean, given the sample data.

In conclusion, determining whether a value falls within a confidence interval is a fundamental skill in statistical analysis. In this case, by calculating the 90% confidence interval for the given sample data, we found that 48.8 is not within the interval (48.3026, 48.6974). This indicates that 48.8 is unlikely to be the true population mean. We've walked through the steps of calculating the margin of error using the z-score for a 90% confidence level and constructing the confidence interval by adding and subtracting the margin of error from the sample mean. Understanding confidence intervals is crucial for making informed decisions based on sample data. It allows us to estimate a range of plausible values for the population mean and assess the compatibility of specific values with the observed data. The confidence interval is a powerful tool for quantifying uncertainty and drawing meaningful conclusions from statistical analyses. Throughout this article, we have emphasized the importance of interpreting confidence intervals correctly. They provide a probabilistic range for the population mean, but they do not represent the probability that the true mean falls within the calculated interval. The confidence level reflects the proportion of intervals constructed from repeated samples that would contain the true mean. By mastering the calculation and interpretation of confidence intervals, you can enhance your ability to analyze data, make informed inferences, and communicate your findings effectively. This skill is valuable across various fields, including research, business, and policy-making. The confidence interval is a cornerstone of statistical inference, and a thorough understanding of its properties and applications is essential for anyone working with data.