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When dealing with sample data, calculating the sample mean is a common task. The sample mean is a measure of central tendency, which shows the average value of a data set. To compute the sample mean, sum all the data values in the sample and divide by the number of observations.
For example, in Sample 1, which contains values {10, 7, 13, 7, 9, 8}, you add them up to get 54, and since there are 6 values in the sample, the sample mean is calculated as \( \bar{x}_1 = \frac{54}{6} = 9 \). Similarly, for Sample 2, the sum of the data values {8, 7, 8, 4, 6, 9} is 42, and the mean is \( \bar{x}_2 = \frac{42}{6} = 7 \).
Sample means provide an estimate of the population mean and help in performing further statistical analyses.

The sample standard deviation quantifies the amount of variation or dispersion in a sample data set. It indicates how much individual data points tend to differ from the sample mean. The formula for sample standard deviation involves subtracting the sample mean from each data point, squaring each result, summing these squares, dividing by \( n-1 \) (where \( n \) is the sample size), and finally taking the square root of that quotient.
In Sample 1, the deviations from the mean \( \ar{x}_1 = 9 \) are {1, -2, 4, -2, 0, -1}, which results in a sum of squares of 26. Dividing by 5 yields a variance of 5.2, and its square root, the standard deviation \( s_1 = \sqrt{5.2} \approx 2.28 \).
Similarly for Sample 2, comparing each point against \( \ar{x}_2 = 7 \) gives deviations {1, 0, 1, -3, -1, 2}, with a variance of 3.2, resulting in a standard deviation \( s_2 = \sqrt{3.2} \approx 1.79 \).
This measure is crucial for computing confidence intervals and hypothesis testing.

In statistical analysis, a point estimate is a single value used to estimate a population parameter. In this exercise, the point estimate is for the difference between the means of two populations. This is achieved by simply subtracting one sample mean from the other.
The formula used is \( \bar{x}_1 - \bar{x}_2 \). For instance, with Sample 1 having a mean of 9 and Sample 2 having a mean of 7, the point estimate of the difference is \( 9 - 7 = 2 \).
This point estimate is a straightforward, yet insightful calculation that provides a simple comparison between the average outcomes of two groups. It acts as the foundation for more complex inferential statistics, like computing confidence intervals.

Understanding the normal distribution is crucial for statistical analysis, especially when estimating population parameters like means. It is a continuous probability distribution that is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Many statistical tests and estimates, including confidence intervals and point estimates, rely on the assumption of normal distribution. This is partly because of its properties, such as the Empirical Rule, which states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
In this exercise, since the samples are taken from normal populations, we use the normal distribution for calculating confidence intervals. For instance, the critical value used (\( z = 1.645 \)) for a 90% confidence interval comes from the standard normal distribution. These characteristics make it vital for making predictions and understanding variability in data sets.