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The **sample mean** is a fundamental statistical concept, especially when estimating population parameters. In simple terms, the sample mean is the average of all the values in your sample. You calculate it by summing up all the data points and then dividing by the number of points.

In the hospital scenario, to find the sample mean of the patients' waiting times, add all the recorded times together and then divide by the number of patients, which in this case is 35.

• The formula for sample mean is:

\[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]

• Where \(\bar{x}\) is the sample mean, \(x_i\) represents each data point, and \(n\) is the number of data points in the sample.

Using the sample mean gives a point estimation for the population mean, although it doesn't reflect the variability or the uncertainty of the estimate, which is why we construct confidence intervals.

The **sample standard deviation** measures the dispersion or spread of data points around the sample mean. It gives insight into the variability within the sample. A higher standard deviation means the data points are spread out over a wider range of values, while a lower one indicates they are clustered close to the mean.

To compute this in the context of the emergency room waiting times:

• Subtract the sample mean from each data point, then square these differences.
• Sum all these squared differences.
• Divide the resulting sum by one less than the total number of data points (n - 1), this adjustment is called Bessel’s correction and it's needed when dealing with samples.
• Finally, take the square root of this result to get the sample standard deviation.

The formula for the sample standard deviation is:\[s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}\]

• Here, \(s\) is the sample standard deviation, \(x_i\) are the data points, \(\bar{x}\) is the sample mean, and \(n\) is the number of data points.

Understanding this concept helps us gauge how much the sample mean might err from the actual population mean.

The **t distribution**, also known as Student's t distribution, is a probability distribution that is symmetric, bell-shaped, and resembles a normal distribution but has thicker tails. This characteristic is crucial because it accounts for the extra uncertainty inherent in small sample sizes.

When constructing confidence intervals for small samples or when the population standard deviation is unknown, the t distribution allows us to determine how much the sample mean is likely to deviate from the true population mean.

The shape of the t distribution changes based on degrees of freedom, which is typically the sample size minus one (n - 1). In our exercise, with 35 patients, we have 34 degrees of freedom.

• You can find the critical value (also known as the t-score) in a t-table or use a calculator, aligning it with a 99% confidence interval and 34 degrees of freedom.
• The t-score helps set the range around the sample mean for the confidence interval.

The t distribution is especially used over normal distribution in situations where the dataset is small, ensuring that the confidence intervals remain accurate under these conditions.

The **population mean** symbolizes the average of a characteristic for an entire population. Unlike the sample mean, which is derived from only a portion of the population, the population mean represents what would occur if you could collect data from every single member of the population.

In our problem, the exact population mean of waiting times in the emergency room is unknown. However, by taking a sample and using it to construct a confidence interval, we can estimate this population mean.

The 99% confidence interval gives us a range believed to contain the true population mean, implying we can be 99% sure that the actual mean falls within this range.

• It is important to remember that increasing the confidence level (say from 95% to 99%) leads to a wider interval, reflecting more certainty but also more potential variability.
• Accurate estimation of the population mean helps in planning and decision making, such as staffing levels in the hospital to reduce wait times.

Thus, comprehension of the population mean is vital for enhancing performance and resource allocation in various practical domains.