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When dealing with statistics, normal distribution is a common concept that plays a key role in understanding data patterns. Imagine it as a bell curve that shows how data points are spread out in a dataset. Most data points tend to cluster around the mean, with fewer points appearing as you move away from it. This symmetry makes it easy to predict outcomes for many real-world scenarios.

In our exercise, the weights of mountain lions are assumed to follow a normal distribution. This assumption is crucial because it allows us to use standard statistical methods, such as calculating confidence intervals, to make inferences about the entire population based on our sample data. By understanding that the weights form a bell-shaped curve, we can estimate the average weight for all mountain lions in the region. This smooth curve also aids in evaluating probabilities and understanding variations within the dataset.

The mean is a measure of central tendency, which helps us understand the average value in a dataset. To calculate the mean, we sum up all the data points and then divide by the number of data points in the set. This gives us an idea of the typical weight of mountain lions in our example.

In our exercise, the mean weight is calculated as follows: Take the weights of the captured lions, add them together, and divide by the total number of weights. Our calculated mean was \(\bar{x} = 96.4 \) pounds. This value gives a snapshot of the central weight around which all others are distributed. However, as noted, our calculation reveals that the provided mean of \(\bar{x} = 91.0 \) pounds was incorrect, showcasing the importance of verifying calculations.

Sample standard deviation is a measurement of how spread out the numbers in a dataset are. It tells us how much the individual weight of each mountain lion deviates from the mean on average. The formula to calculate it takes into account each data point, the mean, and the number of observations.

Here’s how it works: Compute the differences between each weight and the mean, square these differences, sum them up, and then divide by the number of data points minus one. Lastly, take the square root of this quotient. This process gives us a value for the standard deviation, in our case, approximately \(s \approx 30.5 \) pounds. This indicates that while some mountain lions weigh close to the mean, others vary significantly, as evidenced by this fairly large standard deviation number compared to the mean.

The Z-score is a statistical calculation that tells us how many standard deviations a data point is from the mean. It’s especially useful for understanding where a particular data point lies within a normal distribution and is critical for calculating confidence intervals.

In our exercise, we need to determine a 75% confidence interval. This requires finding a Z-score that corresponds to this confidence level from a standard normal distribution table. A 75% confidence level results in a Z-score of approximately 1.15. By plugging this into the confidence interval formula, alongside the mean and standard deviation, we can derive that we expect the true mean weight of mountain lions to fall within a certain range. The Z-score is foundational here because it standardizes our data, allowing us to use these common statistical tools with confidence.