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The sample mean is a simple yet powerful statistical concept. It represents the average of a set of sample values. To calculate the sample mean, you sum all the sample measurements and then divide by the number of samples in the set. In mathematical terms, it is expressed as: \[ \bar{x} = \frac{\Sigma x}{n} \] where \( \Sigma x \) is the sum of all sample values and \( n \) is the number of samples. In the context of our exercise, the sum of the textbook costs (\( \Sigma x \)) was 3582.17, and the number of samples (\( n \)) is 41. Thus, the sample mean \( \bar{x} \) for this set is calculated by dividing 3582.17 by 41, which gives the average cost of a textbook as \( \bar{x} = 87.37 \). This mean gives a centralized point of comparison, helping to understand the general trend in the dataset.

Standard deviation is a metric that indicates the level of spread or variation in a dataset. In simpler terms, it shows how much the sample data points differ from the sample mean. A small standard deviation means that the data points are close to the mean, while a large standard deviation means they are spread out over a wider range. The calculation involves determining the variance first, which is the average of the squared differences from the mean. For a sample, this is calculated as: \[ s^2 = \frac{\Sigma(x - \bar{x})^2}{n - 1} \] where \( \Sigma(x - \bar{x})^2 \) is the sum of the squared deviations from the sample mean, and \( n - 1 \) is the number of samples minus one (also known as the degrees of freedom). With our data, the sum of squared deviations is 9960.336, and \( n - 1 \) is 40. Thus, the variance \( s^2 = 249.009 \). The standard deviation \( s \) is simply the square root of the variance: \[ s = \sqrt{s^2} = 15.78 \]. This value of 15.78 suggests that the costs of textbooks vary moderately around the mean of 87.37.

A confidence interval is a statistical tool used to estimate the range in which a population parameter lies, based on sample statistics. It provides an interval estimate for the population mean and is expressed using a probability known as the confidence level. For a 90% confidence level, there is a 90% chance that the interval will contain the true mean. The formula to calculate a confidence interval is: \[ \bar{x} \pm Z\frac{s}{\sqrt{n}} \] where \( \bar{x} \) is the sample mean, \( Z \) is the Z-score corresponding to the confidence level, \( s \) is the sample standard deviation, and \( n \) is the number of samples. For a 90% confidence interval, the Z-score is approximately 1.645. By inserting our calculated values: - Sample mean \( \bar{x} = 87.37 \) - Standard deviation \( s = 15.78 \) - Sample size \( n = 41 \) The interval is thus \( 87.37 \pm 1.645 \left(\frac{15.78}{\sqrt{41}}\right) \), which simplifies to \( 87.37 \pm 4.06 \). Therefore, the 90% confidence interval is approximately \[ (83.31, 91.43) \]. This range suggests we can be 90% confident that the true average cost of a textbook, for the population, lies between approximately \(83.31 and \)91.43.