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Understanding the normal distribution parameters is crucial when dealing with problems related to the likelihood of certain events, such as the lifetime of light bulbs in our example. The normal distribution is a symmetrical, bell-shaped curve that is defined by two parameters: the mean \( \( \mu \) \) and the standard deviation \( \( \sigma \) \).

The mean or average \( \( \mu \) \) is the center of the distribution, where the highest point of the bell curve lies. It tells us the average value around which the data points are clustered. In our bulb lifetime example, this is 700 hours. The standard deviation \( \( \sigma \) \) indicates how much the individual data points deviate from the mean on average. A smaller standard deviation means that the data points are close to the mean, while a larger one signifies more spread. For the light bulbs, this is 50 hours, showing moderate spread around the mean lifetime.

With these parameters, we can describe the entire distribution of bulb lifetimes and predict probabilities for different lifespans, which enables us to make decisions on maintenance schedules.

The 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. If a z-score is 0, it indicates that the data point's score is identical to the mean score.

In the case of the light bulb example, finding the z-score corresponding to the 20% cumulative probability required us to look it up in a standard normal distribution table or use a calculator with a statistical function. The z-score of approximately \( -0.84 \) suggests that the time by which 20% of the light bulbs fail is 0.84 standard deviations below the mean lifetime of 700 hours.

This value is critical for finding the exact time to replace the bulbs before reaching the 20% failure threshold. By converting a z-score back into raw score using the mean and standard deviation, we get the practical information to make decisions—like when to replace the bulbs.

Cumulative probability refers to the likelihood that a random variable is less than or equal to a specified value. It's a fundamental concept in statistics and probability, often visualized on a graph as the area under the curve to the left of a z-score on the normal distribution curve.

Interpreting the cumulative probability helps in determining how often certain events will occur. For instance, in our light bulb problem, we aimed to determine how long the bulbs should last so that no more than 20% of them would fail. Using cumulative probability, we translated this percentage into a z-score and ultimately into the number of hours after which the bulbs should be replaced.

Cumulative probabilities are not only theoretical but have real-world applications in business, engineering, and other fields, guiding decision-making processes based on statistical evidence. In the light bulb example, using this concept ensures efficient maintenance and cost savings by minimizing premature replacements while avoiding a high frequency of bulb outages.