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Imagine if you charted the times taken by applicants to fill out a mortgage form. The time they take is usually around an average, but there's some randomness in it. Often, this randomness looks like a bell-shaped curve when charted — this is the normal distribution. The peak of this curve represents the average time, here given as 10 minutes.
This distribution is symmetric, meaning it is equally likely for an applicant to take either slightly less or slightly more time than the mean. The normal distribution is crucial because it allows for the calculation of probabilities over an interval or range of values using the parameters of the mean and standard deviation.

• Mean (\( \mu \) )— the center of the distribution, here 10 mins.
• The Standard Deviation (\( \sigma\) ) — measures how spread out the times are, here 2 mins.

Understanding how the data is spread helps predict how likely a particular data point is — useful when determining probabilities like in this problem.

When comparing different data points within a normal distribution, we use something called the Z-score. It tells us how far a data point is from the mean in terms of standard deviations. This is key to understanding probabilities in the normal distribution context.
To find a Z-score:

• Subtract the mean from your data point,
• Divide by the standard deviation of the sampling distribution.

In our exercise, the fixed amount of time we're interested in comparing is 11 minutes. We want to know how far this is from the average of 10 minutes in terms of our sampling distributions. Calculating Z-scores enables us to standardize these comparisons, making it easier to use statistical tables to find the probabilities associated with them.
Once calculated, these Z-scores can be looked up in a Z-table to find the probability of observing a value less than or equal to 11 minutes.

In a real-world scenario, it's not always practical to look at the entire population, so we often take samples. When we calculate means from all possible samples of a given size from a population, these means form a distribution — the sampling distribution.
The distribution of the sample mean will follow a normal distribution even if the source population is not, provided the sample size is large enough. However, with our data being normally distributed even with smaller sample sizes, the sample means will still be normally distributed.

• The mean of this sampling distribution remains the same as the population mean.
• The standard deviation of the sampling distribution is called the standard error, calculated as \( \sigma/\sqrt{n}\).

Sampling distributions help ensure data reliability by giving statistical evidence through probability, which is what we've established in the exercise with five and six applications.

Standard deviation is a critical concept to grasp when dealing with variability in your data. It tells you how much your data points typically differ from the mean.
In the context of our exercise, it's important because it indicates how much the time taken by mortgage applicants varies around the mean of 10 minutes.

• A small standard deviation indicates that most applicants take almost the same time.
• A large standard deviation suggests more variety in filling times.

In the sample means' context, the standard deviation of the sample means is what we refer to as the standard error. It decreases with increasing sample size because the average of a larger sample tends to be closer to the population mean.
Understanding standard deviation and standard error helps in assessing the reliability of our sample mean estimates in representing the population properly.