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The normal distribution is a key concept in statistics that you encounter frequently in probability exercises. It is a continuous probability distribution characterized by its bell-shaped curve, known as the Gaussian curve, which is symmetric around its mean. In the exercise with mortgage applications, the time taken to fill out a form follows a normal distribution with a mean of 10 minutes and a standard deviation of 2 minutes.

A few important properties of normal distribution include:

• The mean (\(\mu\)) determines the center of the curve.
• The standard deviation (\(\sigma\)) measures the spread or width of the distribution.
• Approximately 68% of the data falls within one standard deviation of the mean, while about 95% falls within two.

These properties make the normal distribution useful for modeling various natural phenomena and for making predictions about outcomes, such as the time it takes to complete a task.

In statistics, when we take a sample from a population, we are often interested in understanding the distribution of a sample statistic, like the sample mean. This brings us to the concept of sampling distribution, which is the probability distribution of the sample statistic.

In the given exercise, the sample means for the times taken by five individuals and six individuals to fill out the forms are considered. Each of these sample means has its own distribution, which is centered around the same population mean (10 minutes) but has a smaller spread due to averaging. The standard deviations of these sample means are computed as \(\frac{2}{\sqrt{5}}\) and \(\frac{2}{\sqrt{6}}\) respectively. This reduction in standard deviation as the sample size increases is known as standard error, and it indicates that larger samples provide more reliable estimates of the population mean.

Sampling distributions allow us to make inferences about the population from samples, which is critical in statistical analysis.

The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1. This transformation is very useful because it allows us to use standard normal tables to find probabilities for any normal distribution.

In the mortgage form exercise, to find the probability that the sample average on a certain day is at most 11 minutes, we convert the sample mean to a standard normal variable (\(Z\)) using the transformation: \[Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\]where \(\bar{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size. By converting to a standard normal distribution, we can easily find the probability using standard normal tables or software applications. This makes calculating and understanding normal probabilities much more straightforward.

The sample average, also known as the sample mean, is the sum of a set of observations divided by the number of observations. It serves as an estimate of the population mean and is denoted by \(\bar{X}\).

In the problem given, the sample average is the average time taken by individuals from samples of five and six people to fill out a form. It is a crucial measure because it provides a point estimate of the population mean, which is not always directly accessible.

Key points about the sample average:

• It is affected by sample size—more observations generally lead to a more accurate estimate of the true population mean.
• It reduces variability in the estimation process since it's influenced by the law of large numbers, meaning the larger the sample, the closer the sample average is to the population mean.

Understanding the sample average and its relation to the population mean is foundational in statistics, especially when estimating parameters or testing hypotheses.