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The concept of normal distribution is fundamental in statistics and forms the basis of many statistical analyses. It is commonly represented by a bell-shaped curve that is symmetrical about the mean. For a set of data that follows the normal distribution, the mean, median, and mode are all equal and located at the center of the distribution. The curve's spread is determined by the standard deviation, which measures how much the individual data points deviate from the mean.

When a data set has a normal distribution, as is the case with the women's heights in the exercise, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. In the given exercise, the tall club is interested in the height threshold for the top 2% of women, which is to say the 98th percentile. This value lies well beyond two standard deviations from the mean on the right side of the distribution curve.

A Z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. This score is used to determine how unusual a value is compared to all the values in the data set.

To calculate a Z-score, you subtract the mean from the value in question and then divide the result by the standard deviation of the group. The formula looks like this: \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. In the exercise, the Z-score is used to identify the 98th percentile: it is the number of standard deviations the 98th percentile height is above the average height for women. A positive Z-score indicates a value above the mean, while a negative Z-score indicates a value below the mean. The 98th percentile corresponds to a Z-score of approximately 2.05, meaning that height is a little over two standard deviations above the mean.

Cumulative probability in statistics refers to the likelihood that a random variable takes on a value less than or equal to a specified value. In the context of the normal distribution, cumulative probability is represented by the area under the distribution curve to the left of a given point. This area corresponds to the probability that a value is below a certain threshold and is of chief importance when dealing with percentiles.

For example, to find the 98th percentile in the exercise, we use cumulative probability to determine the smallest height such that 98% of women are at or below that height. The cumulative probability associated with the 98th percentile is 0.98, meaning 98% of the distribution falls below this point. This value is what we look up in the Z-table or calculate using a statistical software to find the matching Z-score. The cumulative probability connects the percentile rank to its equivalent Z-score, thereby enabling us to calculate the corresponding value (the height) in the specific distribution being studied.