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The normal distribution is a fundamental concept in statistics that describes how the values of a variable are distributed. It is often depicted as a bell-shaped curve that is symmetrical around the mean. This is because most of the data points tend to cluster around the mean, with fewer points appearing as you move further away in either direction.
- Mean: This is the central value around which the data is distributed. In the problem, the mean life of a light bulb is 8000 hours.
- Variance: This describes how spread out the data is from the mean.
- Standard deviation: A measure that indicates the extent of deviation for a group as a whole from the mean. In our case, it is 400 hours.
The normal distribution is used frequently because it applies to many natural phenomena. Understanding it allows us to make predictions and probabilities about a population based on a sample.

To understand where a particular value stands within a normal distribution, we use the z-score. A z-score indicates how many standard deviations a data point is from the mean. It's a method of standardizing scores to create comparability.
The formula to compute the z-score is: \( Z = \frac{X - \mu}{\sigma} \), where:

• \(X\) is the data point you're evaluating.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For the given exercise, calculating a z-score allows us to determine what percentage of bulbs fall above or below a certain hour threshold. For instance, a z-score of 0.75 indicates that a data point is 0.75 standard deviations above the mean.

Statistics problems like these often involve finding probabilities related to real-world scenarios, such as light bulb lifespan. By distinguishing percentages in a normal distribution, we can solve for individual values.
The steps include:

• Identifying the given percentage that relates to the cumulative distribution.
• Use of z-tables or calculators to find corresponding z-scores.
• Application of these z-scores to find specific data points within the distribution.

These steps guide us in interpreting data and making informed predictions or decisions based on statistical evidence. For this problem, understanding these methods helped solve for the bulb lifespan at given probabilities efficiently.

Standard deviation is a vital statistical measure representing the amount of variation or dispersion in a data set. A low standard deviation points to data being close to the mean, while a high standard deviation indicates data is more spread out.
It is calculated as the square root of the variance and helps describe how values differ from the mean:
\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2} \]
In the exercise, the standard deviation of 400 hours provides a context of how much the bulb life's hour count typically varies from the mean of 8000 hours. Appreciating this variability is key in assessing the reliability and expectations of the light bulbs.