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The Central Limit Theorem (CLT) is a fundamental concept in statistics. It states that when you have a large enough sample size, the distribution of the sample mean will be approximately normally distributed, even if the original data is not normally distributed. This is powerful because it allows statisticians to make inferences about population parameters using sample statistics.
In this case, even though the original data follows a uniform distribution, the sample mean of these data points will form a normal distribution thanks to the CLT. Here are the key points to remember:

• The CLT applies to sample sizes generally 30 or greater.
• The mean of this new distribution of sample means () will be the same as the population mean .
• The standard deviation of the sample mean, also called the standard error, is the population standard deviation divided by the square root of the sample size (/n).

The application of the CLT allows us to use z-scores and percentiles even with non-normally distributed data. This makes analyzing data more accessible and provides a valid approximation of the sample mean's variability.

The sample mean is a vital statistic that serves as an estimate of the population mean. When you take a sample from a population and calculate the average of the sample, you're essentially calculating the sample mean. It's represented by the symbol \( \bar{x} \) and serves as a point estimate for the population mean, symbolized as \( \mu \). One essential property of the sample mean is that it minimizes the sum of squared deviations from the data points. This means it is the best unbiased estimator for the true population mean.

• Formula for sample mean: \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)
• This is different from a median, which is the middle value, and mode, which is the most frequently occurring value.
• In our context, the uniform distribution over 0 to 75 minutes gives us an expected sample mean of 37.5 minutes.

Calculating the sample mean from a set of data is straightforward and forms the basis of further statistical analysis and inference.

Percentiles are measures that indicate the relative standing of a value within a data set. They are useful for comparing scores or data points across different data sets or against a standard. The nth percentile is the value below which a given percentage, \( n \), of observations in the dataset fall.Let's break down the percentile calculation process, specifically the 90th percentile, as in our example:1. **Locate the Percentile Position**: For percentiles relating to the distribution of a sample mean, this involves using z-scores. - The z-score is a statistic that tells you how many standard deviations a data point is from the mean. - For the 90th percentile, we look up a z-score, which represents 90% of the distribution (1.28 for most tables).2. **Calculate the Percentile Using \( X_{p} = \mu + Z \times \text{Standard Error} \)**: - Find the mean (), and calculate the standard error (standard deviation of the sample mean). - Use the z-score to calculate the actual value  that matches the desired percentile.In our example, we found that the 90th percentile, or the sample mean, corresponds to about 40.3 minutes based on the given values. This process helps practitioners make informed conclusions about the population better, giving valuable insights into the expected range of values.