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Understanding how to calculate the sample mean is fundamental in statistics. The sample mean, denoted as \(\bar{x}\), represents the average value of our sample data. It serves as a point estimate of the population mean \(\mu\) when the population is too large to measure each individual value.

To compute the sample mean, we sum up all the sample data points \(x_i\) and then divide by the number of observations \(n\), which can be expressed as the following formula:
\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\]
This formula suggests that if the data points are equally distributed, then the sample mean would be at the center of the range. In our blood cell count example, the sample mean gives us an idea of the typical red blood cell count for this particular individual over the 15-day period. It's important for students to carefully sum all the values and accurately count the number of observations, ensuring the denominator in the mean calculation reflects the correct sample size.

Remember, in practice, we often use the sample mean as an estimate for the population mean because it's usually impractical to measure the entire population. This assumption is most reliable when the sample is random and representative of the population.

In addition to the mean, the sample standard deviation is essential for understanding the variability or spread of data within a sample. It is denoted by \(s\) and is calculated by taking the square root of the sample variance. The sample variance is the average of the squared differences between each data point and the sample mean.

The formula for the sample standard deviation is:
\[s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2 }\]
The subtracted 1 from the number of observations \(n\) in the denominator is crucial. It corrects the bias in the estimation of the population variance from a sample, providing what is called an 'unbiased estimator'. This process is referred to as 'Bessel's correction'. Students should ensure the use of the sample mean previously calculated as the reference point when subtracting each data value.

Knowing the standard deviation allows researchers to understand how much variation exists from the average, impacting how confidently one can infer about the population from the sample data. In our blood cell count example, the relatively small standard deviation suggests that the measurements were quite consistent across the 15 days.

When we're estimating population parameters, like the mean, from small sample sizes, we turn to the t-distribution, a key concept in inferential statistics. The t-distribution is a probability distribution that accounts for the additional uncertainty introduced when working with a small sample size (generally less than 30). It resembles a normal distribution but has fatter tails, meaning it predicts a higher likelihood of extreme values.

The t-distribution is parameterized by degrees of freedom (df), which for a single sample is \(n-1\). Degrees of freedom relates to the number of independent values that can vary in the calculation of a statistic. In our exercise, the degrees of freedom are 13 since there are 14 observations.

To build a confidence interval using the t-distribution, we find the \(t\)-value that corresponds to our desired level of confidence (e.g., 95%) and df. It is important to use a two-tailed t-distribution for confidence intervals, where 2.5% of the distribution lies in each tail for a 95% confidence level.
\[t_\frac{\alpha}{2}\] is determined using a t-distribution table or software. The t-value is then multiplied by the standard error (the standard deviation divided by the square root of the sample size) to find the margin of error. Combined with the sample mean, this margin defines the range in which we believe the true population mean lies. Understanding the application and interpretation of the t-distribution is essential for statistical analysis, particularly when dealing with small sample sizes or when the population standard deviation is unknown.