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The Z-score is a key concept in statistics that measures how far a data point is from the mean, relative to the standard deviation. It tells us how many standard deviations away a point is from the average value. To calculate the Z-score, use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the data point we are interested in.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

In our exercise, we calculated the Z-score for 697 mm of rainfall. Here is how we found it:

• Mean (\(\mu\)) = 852 mm
• Standard deviation (\(\sigma\)) = 82 mm
• Data point (\(X\)) = 697 mm

Thus, \[ Z = \frac{697 - 852}{82} \approx -1.8902 \]This tells us that 697 mm is approximately 1.89 standard deviations below the mean.

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In the context of a normal distribution, it helps us understand the percentage of data points falling below a particular Z-score.Once we have the Z-score, we can use a Z-table or normal distribution calculator to find the cumulative probability. In the exercise, we found the cumulative probability for a Z-score of -1.8902:- The cumulative probability for \( Z = -1.8902 \) is approximately 0.0294.This means that in approximately 2.94% of all years, the rainfall in India is 697 mm or less. By finding cumulative probabilities, we can understand the likelihood of values within a distribution range.

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. It's a critical component when dealing with normal distribution, as it determines how spread out the values are around the mean. To think about it simply:

• A smaller standard deviation means that the values are closer to the mean.
• A larger standard deviation indicates more spread out values.

In our problem, the standard deviation of the rainfall is 82 mm, which gives us insight into the variability of the monsoon rains. For example, knowing this helps in calculating the Z-scores, which are then used to find probabilities in the normal distribution.

Finding the probability range of a variable means determining the probability of the variable falling between two values within a distribution. This is significant when wanting to identify, for example, what constitutes 'normal' conditions.In the exercise, normal rainfall is defined as between 682 mm and 1022 mm. The probabilities at each bound were calculated using their respective Z-scores:- For 682 mm, the Z-score was \(-2.0732\) with a cumulative probability of approximately 0.0183.- For 1022 mm, the Z-score was \(+2.0732\) with a cumulative probability of approximately 0.9817.The range probability is the difference between these cumulative probabilities:\[ P(682 < X < 1022) = 0.9817 - 0.0183 = 0.9634 \]Thus, in this scenario, there is a 96.34% chance that the rainfall will be considered normal. This helps understand the likelihood of rainfall falling within certain bounds in any given year.