91Ó°ÊÓ

A normal distribution is a continuous probability distribution that is symmetrical around its mean. This bell-shaped curve describes data that cluster around a central point. When we say data is "normally distributed," it means that most of the values lie close to the mean, with frequencies tapering off as you move away in either direction. In our exercise, the concentration of pollutants is normally distributed with a mean of 80 mg/mi. Why does this matter? Because with a normal distribution, we have a predictive model to determine outcomes such as probabilities of data points lying within a specific range. This helps in making precise predictions in various applications, such as assessing pollution levels in a car fleet.

The standard error (SE) measures how much the sample mean of the data is expected to vary from the actual population mean. It tells us about the precision of the sample mean estimate.A lower standard error indicates a more accurate estimate of the true mean. In our exercise, we calculated the standard error by dividing the standard deviation, 4 mg/mi, by the square root of the sample size, which is 25 cars. Hence, the formula used is:\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \]Substituting the given values, we have\[ \text{SE} = \frac{4}{\sqrt{25}} = 0.8 \text{ mg/mi} \] Thus, the standard error helps us understand the variability of the sample means and is crucial for further calculations, such as finding probabilities when using the Z-score.

The Z-score is a statistic that measures how many standard deviations an element is from the mean. The Z-score allows us to determine the position of a data point within a normal distribution and to calculate probabilities for finding values greater or less than a particular point.In our exercise, the Z-score calculation helped us determine the critical level \( L \) such that only 1% of the sample mean of nitrogen oxides exceeds it. We assigned a Z-score corresponding to the 99th percentile of the normal distribution, which is roughly 2.33. This Z-score is then used to find \( L \) through a backward calculation process:\[ L = \mu + Z \times \text{SE} \]Thus, Z-scores are critical for probability calculations when dealing with normal distributions, especially for translating percentiles into specific numeric thresholds.

Probability calculations in a normal distribution help us to figure out the likelihood of an event happening. In simpler terms, it answers questions like "What is the chance?" For example, in the given exercise, we want to find the point beyond which only 1% of the sample means lies.We set up the inequality for probability as \( P(\bar{x} > L) = 0.01 \). This requirement translates into finding the 99th percentile of a normally distributed sample mean. Using the Z-score at this percentile, and implementing the formula:\[ L = \mu + Z \times \text{SE} \]The calculations determine the level \( L \), providing the answer of approximately 81.864 mg/mi for our exercise. By understanding probabilities, we can make informed decisions and predictions, ensuring we are considering the rare events such as the exceedance of pollution levels significantly beyond an expected mean.