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When analyzing data from samples, the **sample mean** is a crucial statistic. It is simply the average of all the data points collected in your sample. In the given exercise, this is the average number of hours U.S. citizens spent watching movies, which is 78 hours per year. To compute the sample mean (\(\bar{x}\)), use the formula:- \[\bar{x} = \frac{\sum{x_i}}{n}\]- where \(x_i\) signifies each data point, and \(n\) is the total number of data points in the sample.The sample mean is used to make inferences about the population mean, aiding in the determination of parameters like the confidence interval. It's a powerful tool that helps summarize a large set of data into a single representative value.

The **standard deviation** quantifies the amount of variation or dispersion in a set of data. In the given example, the standard deviation of 9 hours indicates how much individual movie-watching hours differ from the average 78 hours.The formula to calculate standard deviation (\(s\)) is as follows:- \[s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\]- where \(x_i\) represents each data point, \(\bar{x}\) is the sample mean, and \(n\) is the sample size.A smaller standard deviation implies that data points are close to the mean, while a larger value indicates more spread out data.

The **standard error (SE)** measures how much the sample mean is expected to vary from the true population mean. It provides insight into the precision of the sample mean.To calculate the standard error, use the formula:- \[SE = \frac{s}{\sqrt{n}}\]- where \(s\) is the sample standard deviation, and \(n\) is the sample size.In our exercise, the standard error is approximately 1.27, calculated using a sample size of 50 and a standard deviation of 9. The smaller the standard error, the more reliable the sample mean as an estimate of the population mean.

The **Z-score** is a statistical measure that expresses the number of standard deviations a data point is from the mean. It's pivotal in determining confidence intervals. For confidence interval calculations, a critical Z-score value is used. The exercise mentions Z-scores of 1.96 and 1.645, corresponding to 95% and 90% confidence levels, respectively. - At a 95% confidence level, the Z-score of 1.96 means that the interval captures the true population mean about 95% of the time. - Similarly, a 90% confidence level has a Z-score of 1.645. These values allow us to calculate a range wherein the true population parameter likely falls with a certain level of confidence.

Calculating the needed **sample size** is crucial for determining how many observations are required to make reliable population estimates. It ensures sufficient precision by controlling the margin of error.For the specified margin of error in the exercise, use the formula:- \[n = \left(\frac{Z \times s}{E}\right)^2\]- where \(Z\) is the Z-score for your chosen confidence level, \(s\) is the sample standard deviation, and \(E\) is the desired margin of error.In the example, the calculation determines that a sample size of 220 is required for 90% confidence, ensuring that the sample mean is within 1 hour of the true population mean. Rounding up the calculated value ensures the sample size is realistic and effectively captures the needed data characteristics.