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In probability and statistics, a normal distribution is one of the most commonly used continuous probability distributions. It's often referred to as the bell curve because of its bell-shaped appearance when graphed. A normal distribution is defined by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).- **Mean (\(\mu\))**: This is the peak of the curve, indicating the most likely value of the dataset. In our exercise, this is \(\$21\), which represents the average fare.- **Standard Deviation (\(\sigma\))**: This measures the spread of the distribution, showing how far data points are from the mean. In our case, it's \(3.50\).The usefulness of the normal distribution is in its predictability, as we know that approximately 68% of data will lie within one standard deviation from the mean, 95% within two, and 99.7% within three. This helps in estimating probabilities and making decisions based on statistical data like the taxi fares in our exercise.

The sampling distribution is a critical concept when dealing with samples as opposed to full populations. It refers to the distribution of a statistic (like the sample mean) over many repeated samples from a population. When we take a sample, say of 15 taxi fares, and calculate the mean, that sample mean is one point in the sampling distribution.- **Importance**: Helps in understanding how the sample mean relates to the population mean.- **Properties**: When the sample size grows, the sampling distribution of the sample mean becomes approximately normal due to the Central Limit Theorem (CLT).In our exercise, the sample size is 15. According to the CLT, the distribution of the sample mean will tend towards a normal distribution even if the original fare distribution isn't perfectly normal. This allows us to make predictions about the sample mean as we did by assuming a normal form with a mean of \(\$21\) and a sample mean standard deviation, called the standard error, of approximately \(0.903\).

A Z-score tells us how many standard deviations a data point is from the mean of the distribution. It's a useful measure because it allows us to work with different data units under the same framework of a standard normal distribution.- **Formula**: The Z-score is calculated using the formula: \[Z = \frac{X - \mu}{\sigma_\bar{x}}\] where \(X\) is the value, \(\mu\) is the mean of the distribution, and \(\sigma_\bar{x}\) is the standard error.- **Application in Exercise**: We convert our interval (\(20, 23\)) into Z-scores: - For \(X = 20\): \(Z \approx -1.107\) - For \(X = 23\): \(Z \approx 2.215\)Calculating the Z-scores helps us use standard normal distribution tables or calculators to find probabilities, making interpreting our results easier.

Calculating the probability of a sample mean falling within a certain range involves using Z-scores and the properties of the normal distribution.- **Using Standard Normal Distribution**: Once Z-scores are known, they can be used to find the associated probabilities from a standard normal distribution table or calculator.- **Exercise Application**: We calculated Z-scores for \(\20\) and \(23\) and found the probability that the sample mean is between these two values. For this exercise: - Using these Z-scores (\(-1.107\) and \(2.215\)), we aim to find \(P(-1.107 \leq Z \leq 2.215)\). - This calculation showed that there's approximately 86.5% probability that the sample mean of the fares lies within this range.By mastering probability calculations using the standard normal distribution, we gain a powerful tool for statistical inference, helping us to draw conclusions based on sample data.