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The sample mean is an average that represents the central tendency of a dataset, which helps in understanding the general behavior of the data. To find the sample mean, add up all your data points and then divide by the number of data points. For example, with the numbers 6, 17, 11, 22, and 29 from our exercise:

• Add: 6 + 17 + 11 + 22 + 29 = 85
• Divide by the number of student responses: 5
• Sample mean = 85 / 5 = 17

This result tells us that, on average, these students visited the dentist every 17 months. Calculating the mean is important because it provides a simple number to summarize the group's behavior. It also serves as the starting point for further statistical analysis, like creating confidence intervals.

The sample standard deviation measures how much individual data points differ from the sample mean. It reflects the variability in the dataset. To find the sample standard deviation, follow these steps:

• Subtract the sample mean from each data point:
  • (6 - 17), (17 - 17), (11 - 17), (22 - 17), (29 - 17)
  • Results in: -11, 0, -6, 5, 12
• Square each result:
  • 121, 0, 36, 25, 144
• Sum these squared differences:
  • 121 + 0 + 36 + 25 + 144 = 326
• Divide by the number of data points minus one (n-1):
  • 326 / 4 = 81.5
• Take the square root of this result:
  • \ \( \sqrt{81.5} \approx 9.03 \)

The standard deviation of approximately 9.03 tells us that the months since the last dentist visit varied by about 9 months from the average. A larger standard deviation means more variability, which can affect the width of confidence intervals.

The t-distribution is a curve that is used when dealing with small sample sizes, usually less than 30. It resembles the normal distribution but has thicker tails, accommodating for variability in smaller datasets. It's particularly used for calculating confidence intervals or conducting hypothesis tests when the population variance is unknown.When constructing a confidence interval with a t-distribution, we need the t-value, which depends on:

• The confidence level (commonly 95%)
• The degrees of freedom, calculated as the number of data points minus one (n-1)

For our example with 5 students, the degrees of freedom are 4. Using a t-distribution table, or a calculator, the t-value for a 95% confidence level is approximately 2.776.This t-value adjusts the confidence interval width to account for small sample sizes. The confidence interval formula is:\[ x̄ \pm t \left( \frac{s}{\sqrt{n}} \right) \]where:

• \( xÌ„ \) is the sample mean
• t is the t-value
• s is the sample standard deviation
• n is the number of data points

This approach gives us a reliable way to infer the average behavior of a larger population based on a small sample.