91Ó°ÊÓ

The sample mean is a fundamental concept in statistics. It represents the average value of a set of data, and is calculated by adding together all the individual data points, then dividing by the number of data points in the set. In our example, we have the red blood cell counts recorded over 15 days. To find the sample mean for these data, all the individual counts are summed up, and then this total is divided by 15, the number of days. For these counts, the sample mean is approximately 5.2533 million cells per microliter. The sample mean gives us a central value around which the data points are distributed. It provides a concise summary of the data set. This value is especially significant when constructing a confidence interval, as it represents our best estimate for the true population mean, based on the sample data we have.

Sample standard deviation is a measure of the spread or dispersion of a set of data points. It tells us how much the individual data points deviate, on average, from the sample mean. Calculating the sample standard deviation involves finding the square root of the average squared deviations. In our calculation, after determining the sample mean, each red blood cell count is subtracted from the sample mean, and each result is squared. These squared differences are then averaged by dividing by one less than the number of data points (14 in this case, since it's a sample), and finally taking the square root to give our standard deviation. For this sample, the calculated standard deviation is approximately 0.1532 million cells per microliter. This value provides insight into how much variation or "scatter" exists in the blood cell counts around the sample mean, giving us a clearer understanding of the data's consistency.

The t-distribution is a statistical distribution used when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but with thicker tails, which accounts for the extra variability introduced by estimating the population standard deviation from a small sample size. In the context of our exercise, we use the t-distribution because we are working with a sample size of 15, which is relatively small. With 14 degrees of freedom (one less than the sample size), we refer to a t-distribution table or calculator to find the critical t-score for a 95% confidence level. The relevant t-score is approximately 2.145. This t-score helps us construct the interval around the sample mean, reflecting the uncertainty in our estimate and thus appropriately accounting for variation when making inferences about the population mean.

The margin of error is a statistic that quantifies the uncertainty in a sample statistic's estimate of a population parameter. It essentially sets the range above and below the sample statistic within which the true population parameter is likely to lie.For the red blood cell count data, the margin of error is calculated using the critical t-score from the t-distribution, the sample standard deviation, and the square root of the sample size. The formula for calculating the margin of error is: \[E = t \times \frac{s}{\sqrt{n}}\]where "t" is the critical t-score, "s" is the sample standard deviation, and "n" is the sample size. For our data, this yields a margin of error of approximately 0.0900.The margin of error is added to and subtracted from the sample mean to form the confidence interval, providing a range within which we can say with confidence that the true mean red blood cell count lies.