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The sample mean is an essential measure in statistics that helps us understand the central tendency of a data set. It provides a quick summary of where the data points tend to cluster. To compute the sample mean, sum up all the observations in your sample and divide by the total number of observations. It's expressed as \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) are your data points and \( n \) is the number of observations. In our exercise, the sum of child-care costs was 986, resulting in a sample mean of 98.6 when divided by the 10 observations. This value represents the average amount spent by the employees on child care, offering a base for further statistical analysis.

The standard deviation is crucial for understanding data variability. It quantifies how much the individual data points deviate from the sample mean. A larger standard deviation indicates that the data points are spread out over a wider range, whereas a smaller one suggests that they are closer to the mean. To calculate the standard deviation for a sample, use the formula: \[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \] Here, \( x_i \) represents each data point, and \( \bar{x} \) is the sample mean. The subtraction from each data point to the mean, then squaring, helps in identifying the spread of the dataset. The results are summed up and divided by \( n-1 \) (i.e., degrees of freedom), and finally, the square root is taken. Our calculated standard deviation was approximately 5.54, indicating a moderate level of variability in child-care costs among the employees.

The standard error of the mean (SEM) quantifies the precision of the sample mean as an estimate of the true population mean. It tells us how much the sample mean is expected to vary from the actual population mean if we were to take multiple samples. The SEM is calculated by dividing the standard deviation by the square root of the sample size:\[ SEM = \frac{s}{\sqrt{n}} \] In our example, with a sample size \( n = 10 \) and standard deviation \( s = 5.54 \), the SEM is calculated to be approximately 1.75. A smaller SEM indicates that the sample mean is a more accurate reflection of the population mean, suggesting relatively reliable results from our sample.

When we work with smaller samples, such as 10 in our example, we use the t-distribution instead of the normal distribution to calculate confidence intervals. The t-distribution adjusts for the sample size, ensuring more accuracy when estimating the population mean. It requires knowing the degrees of freedom, calculated as \( n-1 \). With 9 degrees of freedom and a 90% confidence level, we can find the t-critical value using a t-distribution table.For our example, the t-critical value was approximately 1.833. This factor helps adjust the margin of error when constructing the confidence interval. The t-distribution is particularly useful for smaller datasets, offering a safety net by accounting for more variability than the typical normal distribution.