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The sample mean is a critical concept in descriptive statistics. It represents the average of a set of observations. In this context, it is the average number of hours the surveyed Americans watched television each month. Calculating the sample mean is straightforward; you sum up all the individual observations and then divide by the number of observations.

Mathematically, the sample mean is represented as \( \overline{x} \) and is calculated using the formula:

• \( \overline{x} = \frac{\sum_{i=1}^{n} x_i}{n} \)

where \( x_i \) are the individual observed values and \( n \) is the sample size.

The sample mean provides a snapshot of the central tendency of the data. In our example, this mean is 151 hours, indicating that, on average, each participant watched television for 151 hours in a month.

The sample standard deviation, symbolized by \( s_{x} \), measures how much the values in a dataset deviate from the mean. It provides insight into the spread or variability in the data. A smaller standard deviation means that the values are closely clustered around the mean, whereas a larger value indicates a wider spread.

The formula for sample standard deviation is:

• \( s_{x} = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2 } \)

This formula accounts for every data point's deviation from the mean and averages these deviations over \( n-1 \) instead of \( n \), which corrects the bias in the estimation of a population standard deviation.

In our example, \( s_{x} = 32 \) hours, meaning the viewing hours per month deviate from the average by roughly 32 hours.

Degrees of freedom are an important concept in statistical calculations, especially in the context of variance and standard deviation. They refer to the number of independent values or quantities which can vary in the calculation of a statistic.

For the calculation of the standard deviation or variance, the degrees of freedom are given by \( n-1 \), where \( n \) is the sample size. This adjustment is necessary to give us an unbiased estimate when calculating the sample variance.

• \( \text{Degrees of Freedom} = n - 1 \)

In our scenario, with a sample size of 108, the degrees of freedom are 107. This means that while 108 data points are available, only 107 contribute independently to the calculation of variance and standard deviation.

The sample size, denoted by \( n \), is the number of observations or data points collected in a study. It is a fundamental concept as it affects the reliability of the statistical results. Larger sample sizes generally lead to more reliable and precise estimates of the population parameters.

In our case, the sample size is 108, meaning 108 Americans were surveyed. This value is crucial because it directly influences the accuracy of the sample mean and the standard deviation calculations.

• Larger sample sizes reduce variance and increase the confidence in the results.
• Small sample sizes might not adequately represent the population, leading to less reliable conclusions.

Thus, when examining statistical results, always consider the effect of the sample size on the findings.