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In a normal distribution, the mean and standard deviation are essential parameters that describe the dataset. The mean \((\mu)\) is the average of all data points, providing a central value around which the data is distributed. In our exercise, the mean is 0.4 seconds, indicating that most drivers have a reaction time close to this value.
The standard deviation \((\sigma)\) measures how much the data varies from the mean. A smaller standard deviation signifies that the data points are closely clustered around the mean, while a larger one indicates more spread out data. In this exercise, the standard deviation is 0.05 seconds, suggesting that reaction times are not widely spread from the mean.
Together, the mean and standard deviation define the shape of the normal distribution: a symmetrical bell curve centered around the mean, where the spread of the curve is determined by the standard deviation.

Calculating probabilities in a normal distribution involves determining the likelihood that a data point falls within a certain range. This can be done using the standard normal distribution table, also known as the Z-table.
For example, to find the probability of a reaction time greater than 0.5 seconds, we convert the time to a z-score. With a mean of 0.4 seconds and a standard deviation of 0.05 seconds, the z-score calculation for 0.5 seconds results in \( z = 2 \).
The Z-table shows the cumulative probability up to that z-score, implying the probability of being less than your calculated z-score. Thus, to find the probability of exceeding it, subtract the cumulative probability from 1. In this case, for \( z > 2 \), the cumulative probability for \( z < 2 \) is 0.9772, meaning that the probability for \( z > 2 \) is \( 0.0228 \) or 2.28%.
Similarly, to find the probability of a reaction time between 0.4 and 0.5 seconds, calculate the probability between their corresponding z-scores, which are 0 and 2 respectively. From the Z-table, subtract the cumulative probability of the lower z-score from the higher one, resulting in a probability of 47.72% for this interval.

The z-score is a statistical measurement that describes a value's relationship to the mean in a set of values. A z-score indicates how many standard deviations an element is from the mean.
To compute a z-score, use the formula: \[ z = \frac{x - \mu}{\sigma} \]where \( x \) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
In the context of our exercise, if you want a specific reaction time \( x \) that only 10% of drivers exceed (part c), you need to locate a z-score corresponding to a cumulative probability of 90% in the Z-table. This gives \( z = 1.28 \).
You then use the formula to find the reaction time:\[ x = \mu + z \cdot \sigma \]Substituting the known values gives \( x = 0.4 + 1.28 \times 0.05 = 0.464 \) seconds. Therefore, the reaction time of 0.464 seconds is exceeded by 10% of drivers, illustrating the power and flexibility of the z-score in probability calculations and data analysis.