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To understand probability in the context of a normal distribution, we make use of the Z-score. The Z-score is a way to standardize a value within a data set to see how it compares to the mean. By converting a score into a Z-score, you are finding out how many standard deviations an original score is from the mean.The formula to calculate the Z-score is: \[ Z = \frac{X - \mu}{\sigma} \] where:

• \(X\) is the original data point,
• \(\mu\) is the mean of the data set,
• \(\sigma\) is the standard deviation.

Using the Z-score, we can translate an individual data point within a normal distribution to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This makes it easier to calculate the probability of a data point occurring within a certain range. For instance, for the heights of 18-year-old men, you would convert the specific heights (like 67 or 69 inches) into Z-scores to find where they fall in a standard normal distribution.

Standard deviation is a measure of the spread, or variance, of a set of data. Within a normal distribution, the standard deviation tells us how much the individual data points differ from the mean of the data set. The formula for calculating standard deviation in a population is: \[ \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \] where:

• \(\sigma\) is the standard deviation,
• \(X\) represents each value in the data set,
• \(\mu\) is the mean of the data set,
• \(N\) is the number of observations.

Standard deviation is key in determining how spread out the data points are in a normally distributed data set. In this exercise, the standard deviation of the heights of 18-year-old men is 3 inches. This means most individuals' heights will fall within 3 inches above or below the mean height of 68 inches. It is important in calculating the probability as it defines the width of the normal distribution curve.

After determining the Z-scores, we then calculate the probability that a value falls within a certain range by considering these scores. This is done using the standard normal distribution table or a calculator that can interpret Z-scores:- **Interpreting Z-scores:** For the heights problem, once you've found the Z-scores for the range (67 and 69 inches converted to -0.333 and 0.333), use the table to find the probability of values falling between these scores.- **Calculating probability:** For a single value, you would find the area under the normal curve between your calculated Z-scores. Usually, these standardized tables give you the cumulative aggregate from the mean to the Z-score, allowing you to ascertain the probability of a value falling within your range.When dealing with a sample mean, like in part (b) of the exercise, you must consider the standard error (SE), which uses the formula:\[ SE = \frac{\sigma}{\sqrt{n}} \] where \(n\) is the sample size.The probability for a group, calculated using the standard error, usually results in higher values due to reduced variance in sample means. For example, finding \(P(-1 < Z < 1)\) doesn't only depend on individual height, but rather on the distribution of the sample mean, making the range tighter and, therefore, the probability higher.