91Ó°ÊÓ

The population standard deviation is a measure of how spread out the values in a full population are around the mean. In the context of the lettuce weights, it tells us how much the weight of all the lettuce heads varies from the mean weight. Unlike the sample standard deviation, which is calculated from a subset of the population, the population standard deviation considers every single data point in the population. This adds accuracy to the measurement of variability because it uses the actual values of the entire group, rather than estimates from a sample. Knowing the population standard deviation allows us to confidently apply certain statistical methods, such as the normal distribution, under specific conditions. In this exercise, it is given as 0.2 pounds, providing a foundational parameter to apply our statistical approach.

Sample size is the number of observations or data points collected from a population for analysis. In our lettuce example, the sample size is 20, which means 20 heads of lettuce were weighed. Why is sample size important? It greatly influences the precision of our estimates and the power of our statistical tests. Larger sample sizes generally provide more reliable data estimates, leading to more robust conclusions about the entire population. However, even with a smaller sample size of 20, we can still use the normal distribution to analyze this data because the population standard deviation is known, and the population itself is normally distributed. This allows us to bypass the rule of needing a sample size of 30 or more when certain conditions about the population are met.

A normal population distribution is a probability distribution that is symmetric about the mean, where most of the observations cluster around the central peak and the probabilities for values taper off equally on both sides. Often referred to as a bell curve, it plays a significant role in statistics. Whenever a population distribution is described as normal, like our exercise with the heads of lettuce, it implies:

• The mean, median, and mode of the distribution are equal.
• Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations (known as the empirical rule).

Understanding that a population distribution is normal allows statisticians to make predictions and inferences more confidently. In our case, since the lettuce weights follow a normal distribution, and we know the population standard deviation, the normal distribution framework becomes the correct choice for analyzing our sample data.