7.2.2 Estimating a Population Mean - Sample Mean, Margin of Error, and Finding Confidence Intervals

This paragraph introduces the concept of estimating the population mean using a point estimate and a confidence interval. It emphasizes that the sample mean (x-bar) is the best point estimate for the population mean due to its properties as an unbiased estimator. The paragraph also explains the process of calculating the margin of error for a confidence interval when the population standard deviation is unknown, highlighting the importance of the sample size and the confidence level in determining the error. The formulas for the confidence interval are reiterated, and the concept of the sampling distribution of sample means is introduced as the basis for the confidence interval calculation.

The second paragraph delves deeper into why the sample mean is considered the best point estimate of the population mean. It discusses the unbiased nature of the sample mean, meaning that its expected value is equal to the population mean. The paragraph also touches on the concept of consistency, which refers to the sample mean's reduced variability compared to other measures of central tendency. The summary explains that a consistent estimator has a smaller standard deviation in its sampling distribution, making it a more reliable estimator as the sample size increases.

This paragraph focuses on the interpretation of the margin of error within the context of constructing a confidence interval for population means. It describes the process of determining the most likely values of the sample mean (x-bar) and how these values relate to the population mean. The explanation includes rearranging the inequality to place the population mean in the center, which allows for the calculation of the confidence interval. The paragraph also clarifies the meaning of confidence levels and how they relate to the probability that the true population mean falls within the calculated interval.

The fourth paragraph outlines the prerequisites for constructing a confidence interval when the population standard deviation is unknown. It stresses the importance of having a simple random sample and ensuring that the sampling distribution of the sample mean is approximately normal, which can be achieved if the original population is normally distributed or if the sample size is large enough (n > 30). The paragraph also details the steps involved in computing the confidence interval, including identifying the necessary components such as the sample mean, sample standard deviation, sample size, and degrees of freedom.

In this paragraph, an example is provided to illustrate the process of constructing a 95% confidence interval using a set of numerical data representing DNA base codes. The example guides through checking the requirements for constructing the interval, computing the sample mean and standard deviation, determining the critical t-value, and calculating the margin of error. It concludes with the calculation of the upper and lower bounds of the confidence interval and a brief discussion on the practical use of the interval in the context of the data provided.

The final paragraph reflects on the practical application of the confidence interval constructed in the example. It points out that while the exercise was valuable for practicing the method, the actual data used (DNA base codes) does not lend itself to meaningful statistical analysis in this context. The paragraph concludes by emphasizing the importance of understanding when and how to apply statistical techniques appropriately, acknowledging that the confidence interval, in this case, has no practical use due to the nature of the data.