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The t-score is a value derived from the Student's t-distribution, which is used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It helps in determining how far the sample mean is from the population mean in terms of standard errors. When the sample size is less than 30, and especially when the population standard deviation is not available, we rely on the t-score rather than the z-score.

• We calculate the t-score by looking up values in a t-distribution table, which is organized by levels of confidence and degrees of freedom.
• The table gives us the critical value of t for our specific confidence level and degrees of freedom.

In our example, with a sample size of 24, the degrees of freedom is 23, and for a 95% confidence level, the t-score is approximately 2.069.

The sample mean (\(ar{x}\)) represents the average value of a sample, which is a subset of the total population. It provides an estimate of the population mean (denoted as \(\mu\)). Measuring the sample mean is crucial because it is used in various statistical calculations, such as calculating the confidence interval, standard error, and more.
To compute the sample mean, you add up all the sample data points and divide by the number of data points: \[\bar{x} = \frac{(sum\ of\ sample\ values)}{(sample\ size)}\] In the exercise, the sample mean is given as 16.7. This value is central to constructing our confidence interval as it is the midpoint around which we calculate the range of possible values for the population mean.

Standard deviation (\(s\)) is a measure that describes the spread or variability of a dataset. In other words, it tells us how much the individual data points in a sample deviate from the sample mean. A low standard deviation means that most of the numbers are close to the average, while a high standard deviation indicates a wider spread of values.

• To calculate the sample standard deviation, find the differences between each data point and the mean, square these differences, find the average of these squared differences, and take the square root of that average.

In the given problem, the sample standard deviation is provided as 2.6. This measure is crucial for calculating the standard error and, subsequently, the confidence interval, as it gives an indication of how much the sample mean might vary from the true population mean.

Degrees of freedom (\(df\)) are linked to the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the context of confidence intervals and t-scores, degrees of freedom are primarily used to determine the appropriate critical t-score from the t-distribution table.

• For a sample, the degrees of freedom are calculated as the sample size minus one: \(n - 1\).
• The reason we subtract one is to account for the use of the sample mean in our calculations, which imposes a constraint on variability.

In our exercise, with 24 samples, the degrees of freedom is 23. This is an essential value, as it determines the specific t-score used to calculate the confidence interval, ensuring accurate statistical inference from the sample to the whole population.