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A Z-score is a way to tell how far a particular data point is from the average, in terms of the number of standard deviations. The formula to calculate a Z-score is \[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value we are examining,
• \(\mu\) is the mean of the distribution, and
• \(\sigma\) is the standard deviation.

Using Z-scores, we can determine how unusual a particular measurement is within the context of a normal distribution. A Z-score of 0 indicates the data point is exactly at the mean, while positive or negative Z-scores indicate the data point is above or below the mean, respectively. For example, if India received 697 mm of rain, we can calculate its Z-score to understand how it compares to the typical rainfall. In this case, the Z-score was -1.89, indicating that 697 mm of rain is 1.89 standard deviations below the average of 852 mm. Understanding Z-scores helps us quantify unusual occurrences, like droughts, by showing how far they deviate from what is typical.

Probability is the measure of the likelihood that an event will occur. In the context of the normal distribution, probability helps us understand the chances of a particular outcome, such as receiving a specific amount of rainfall in a year. It is generally expressed as a number between 0 and 1, with 1 indicating certainty and 0 indicating impossibility.

To find the probability of receiving 697 mm or less of rain, we can use the Z-score obtained from our calculation. Since a Z-score translates an original value into how many standard deviations away it is from the mean, we can use this to look up probability in a standard normal distribution table. The probability of a Z-score of -1.89 or less (697 mm or less of rain) is approximately 0.0294. This means there is a 2.94% chance of receiving 697 mm or less rain in any given year.
Remember, probability helps us understand how likely an event or given outcome is within a certain set of parameters, like the range of rainfall over a series of years.

Cumulative probability is the likelihood that a random variable is less than or equal to a specific value. This helps in situations where we wish to determine the probability of not just a single occurrence but any occurrence up to a certain point.

In the exercise, cumulative probability is used to determine the likelihood of having 697 mm or less rainfall in India during a monsoon season. After calculating the Z-score of -1.89 for 697 mm, we consult the standard normal distribution table to find the cumulative probability, which is 0.0294. This indicates there is a 2.94% chance of experiencing such low rainfall or less.

Cumulative probability is also useful when determining the percentage of normal rainfall years. By finding the Z-scores for both 682 mm and 1022 mm (the lower and upper bounds of what is considered normal), and then using their respective cumulative probabilities, we can calculate the probability of receiving rainfall within that range. The difference between the cumulative probabilities of the upper and lower bounds (98.08% - 1.92%) gives us the cumulative probability for normal rainfall, which turns out to be 96.16%.

• This approach helps encapsulate the overall probability for a range of values in a distribution, making it a vital concept in statistical analysis.