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The standard deviation is a key concept in understanding the normal distribution. It measures how much individual data points in a dataset deviate from the mean. In simple terms, it tells us how spread out the values are. A larger standard deviation means the data points are more spread out from the mean, while a smaller standard deviation indicates they are closer to the mean.

In our scenario with the automobile batteries, the standard deviation of 80 days means that the life spans of the batteries vary by an average of 80 days from the mean of 1110 days. This value provides a sense of consistency or variability in the battery life span. It is essential because it helps to quantify the amount of variation or dispersion in a set of values.

The standard error is a statistical term that provides an estimate of the standard deviation of a sampling distribution. It is crucial when we are dealing with sample data rather than population data.

In the exercise, we use the formula to find the standard error:\[\text{Standard Error} (SE) = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size. For our 400 batteries, the standard error is 4 days. This means that the spread or variability of the average battery life is smaller with a larger sample size. Thus, it gives us a clearer and more accurate picture of where the true mean might lie.

A z-score is a measure that describes the position of a raw score in terms of its distance from the mean, measured in standard deviation units. Z-scores are a way to standardize different datasets, allowing for direct comparison.

In our battery example, to find out how far a certain sample mean is from the overall mean of 1110 days, we compute the z-score as:\[z = \frac{\text{sample mean} - \mu}{\text{SE}}\]The z-score helps us determine the probability of a particular data point occurring within the context of the normal distribution. For instance, a z-score of 0 means the sample mean is exactly at the population mean, while a z-score of 2.5 suggests it is 2.5 standard errors above the mean. This standardization enables us to utilize standard normal distribution tables to calculate probabilities effectively.

Probability calculation within the context of a normal distribution allows us to determine the likelihood of certain outcomes. In our exercise, we are interested in probabilities for different ranges of the average battery life.

To calculate probabilities, we've used z-scores. Once we have a z-score, we look up the probability in a standard normal distribution table or use a calculator. This probability shows us how often we can expect a particular outcome based on our model and assumptions. For example, a z-score for 1100 gives us the probability of a sample mean being out this far from the mean, which we calculated as approximately 49.38%. This indicates there's a high chance of the average falling within this range. Similarly, low probabilities for extremely high or low averages underscore the rarity of such events, as seen with probabilities for averages greater than 1120 or less than 900, calculating to be just 0.62% and virtually 0%, respectively.