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The sample mean is a statistical measure used to summarize the central tendency of a dataset. It is particularly useful when you're looking at a specific group or sample from a larger population, helping you make inferences about the population as a whole.
In our example, the sample mean of the profit per car sold was calculated from data collected from 20 dealers. It came out to be 290, which is an average that represents the typical profit made per car in this sample.
Calculating the sample mean involves summing up all the values in your sample data and dividing by the total number of observations, called the sample size. This is expressed in a formula as:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where:

• \( \bar{x} \) is the sample mean,
• \( \sum x_i \) is the sum of all observed values,
• \( n \) is the sample size.

It's important because it gives a single value that describes the distribution of data points from your sample.

The t-distribution is a probabilistic distribution used in statistics to model sample data, especially useful when dealing with smaller sample sizes (less than 30). It's similar in shape to the normal distribution but has thicker tails, which means it accounts for higher variability when the sample size is small.
In our scenario, since the sample size is 20, we use the t-distribution to determine the appropriate t-value when calculating the confidence interval. This distribution helps us account for the additional uncertainty due to the smaller sample size.
The t-distribution is characterized by its degrees of freedom (df), where df = n - 1. Here, n represents the sample size. So for our 20 dealers, df is 19. Using the t-distribution table, we find that at a 95% confidence level, the t-value is approximately 2.093.

The standard error (SE) is a measurement that describes the extent to which the sample mean is likely to vary from the actual population mean. It's an essential component in the calculation of confidence intervals.
The standard error decreases as the sample size increases, meaning larger samples generally give more reliable estimates of the population mean.
In our case, the standard error is calculated using the formula:
\[ SE = \frac{s}{\sqrt{n}} \]
where:

• \( s \) is the standard deviation of the sample,
• \( n \) is the sample size.

With a sample standard deviation of 125 and a sample size of 20, the standard error is approximately 27.95. This number helps determine the width of the confidence interval and reflects the accuracy of your sample mean as an estimate of the population mean.

Degrees of freedom (df) is a concept used to describe the number of independent values or quantities that can be assigned to a statistical distribution. In the context of a t-distribution, degrees of freedom are essential in determining the shape of the distribution and the specific t-value needed for confidence interval calculations.
For our purpose of calculating confidence intervals, the degrees of freedom are calculated as:
\[ df = n - 1 \]
where \( n \) is the sample size.
In the example with 20 dealers, the degrees of freedom would be 19. This figure allows us to look up the corresponding t-value in a t-table, which helps us account for sample variability in our analysis. Understanding degrees of freedom helps assure that results are adjusted for sample size and thus provides more accurate statistical interpretations.