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The normal distribution is a key concept in statistics, often referred to as the "bell curve" due to its shape. It is symmetrical around the mean, meaning that the left and right sides of the graph are mirror images. This distribution plays a crucial role in statistics because it allows statisticians to make inferences about populations based on sample data. However, in the scenario provided by our exercise, where 95% of the customers did not make any orders (resulting in $0 sales), the data set is heavily skewed towards zero. This skewness indicates that the distribution of the data is far from normal.

The assumption of normality is significant as it impacts the validity of statistical procedures, such as calculating confidence intervals. If the data does not follow a normal distribution, standard methods for confidence interval estimation may not be valid or may require modifications. When working with non-normally distributed data, alternative methods or adjustments might be necessary to obtain accurate statistical inferences.

The sample mean is a measure of the central tendency of a data set, calculated as the sum of all observed values divided by the number of observations. In our exercise, the sample mean is given as $10. This value tells us the average order amount across all customers sampled.

The sample mean serves as an unbiased estimator of the population mean, especially in large samples. This means that, on average, the sample mean will equal the population mean if you take many samples. However, with the heavy skewness observed in the data in the exercise, the sample mean alone may not be a fully adequate summary without considering its variability and the shape of the distribution. In contexts like this, complementing the sample mean with additional analyses or descriptive statistics can provide a more comprehensive understanding of the dataset.

The standard deviation is another critical concept in statistical analysis. It measures the amount of variation or dispersion in a set of values. A low standard deviation indicates the data points are close to the mean, while a high standard deviation suggests they are spread out over a broader range.

In our scenario, the standard deviation is also $10. This value, combined with the sample mean, gives insight into how our sales values deviate from the center point. Even when data is not normally distributed, as is the case here, the standard deviation still acts as a helpful measure of spread. It's important to understand that while the mean tells us where the center of our data is, the standard deviation gives us a picture of how varied the data is around that center, offering a fuller understanding of data variability.

In statistics, the t-distribution is used to estimate population parameters when the sample size is small, or when the population standard deviation is unknown. It is similar to the normal distribution but has thicker tails. This means there is more probability in the tail ends, which accounts for greater uncertainty in estimates from small samples.

In the exercise, we encounter the t-distribution because we don't know the true population standard deviation, and although we have a large sample size (n=100), the heavily skewed data calls for caution. The t-distribution is utilized to derive a 95% confidence interval for the mean under the given conditions. This adjustment helps to achieve reliable estimates even if the data doesn't perfectly adhere to normal distribution assumptions. The critical t-value used (approximately 1.984 for 99 degrees of freedom) accounts for sampling variability and provides an interval thought to include the true population mean with 95% confidence.