91Ó°ÊÓ

The normal distribution is a fundamental concept in statistics that describes how the values of a variable are distributed. It is symmetrical, with most values clustering around the mean and fewer occurring as you move away from the mean. Imagine a bell-shaped curve when visualizing a normal distribution; this curve represents the probability distribution of a continuous random variable. In the context of our exercise regarding women's heights, we assume that the heights are normally distributed around the mean, which means that most women's heights are close to the average with progressively fewer women being much taller or shorter.

In practical terms, if you were to plot the heights of a large number of women on a graph, you would see most data points hovering around the center with the frequency tapering off as the heights deviate from the mean. This pattern is central to many natural phenomena and is used in statistical methods to make inferences about populations from samples.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value), while a high standard deviation indicates that the values are spread out over a wider range. In the scenario of women's heights, a standard deviation of 2.5 inches means that the heights vary by this amount on average from the mean height of 65 inches. This value helps us understand how much individual women's heights can differ from the average height.

Understanding standard deviation is crucial when looking at samples from a population because it allows you to assess how representative your sample is. For instance, if you have a sample with a very different standard deviation than that of the population, it might indicate that your sample is not very representative.

A z-score, or standard score, is the number of standard deviations an element is from the mean. By converting values to z-scores, you can compare them to a standard normal distribution regardless of the original distribution's shape or scale. To calculate a z-score, you subtract the mean from the value in question and then divide by the standard deviation. This process is as shown in Step 1 of the exercise solution.

In our height example, calculating a z-score for a woman 63 inches tall allows us to assess how unusual this height is within the context of the normal distribution of women's heights. A negative z-score indicates a value below the mean, while a positive z-score signifies a value above the mean. As per the provided steps, once the z-score is found, we refer to the standard normal distribution table to find the probability associated with that z-score.

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 represents a certain event. In the context of the exercise, probability is used to determine how likely it is that a randomly selected woman is below a certain height or that the mean height of a sample of women is below a certain height.

Once a z-score is calculated, finding the corresponding probability involves using a standard normal table, or z-table, which lists the probabilities associated with various z-scores. These probabilities tell us the chance of the variable falling to the left of the z-score on the bell curve of the normal distribution. For example, if the z-score for height is found to be -0.8, we can look up this score in the z-table to find the probability of a randomly selected woman being below that height.

The sample mean is the average value of a sample and is calculated by summing up the measurements of all the sample members and dividing by the number of members in the sample. When dealing with probability and the normal distribution, we often use the sample mean to represent the central tendency of the sample data.

The Central Limit Theorem is a statistical theory that states that the sampling distribution of the sample means approaches a normal distribution as the sample size grows, regardless of the population's distribution shape. In Steps 2 and 3 of the provided solution, we use this principle to find the probability that the mean height of a sample is less than 63 inches—first for a small sample size of 5 and then for a larger one of 30. As the sample size increases, the sampling distribution of the mean becomes increasingly normal.