Question

Gender in the Rock and Roll Hall of Fame From its founding through \(2015,\) the Rock and Roll Hall of Fame has inducted 303 groups or individuals. Forty-seven of the inductees have been female or have included female members. \(^{20}\) The full dataset is available in RockandRoll. (a) What proportion of inductees have been female or have included female members? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion female or with female members for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

Short answer

Approximately 15.5% of the inductees have been female or have included female members. The distribution of sample proportions is expected to be normally distributed and centered around 0.155.

Step-by-step solution

Calculate the Proportion

The proportion is calculated by dividing the number of female inductees or groups with female members by the total number of inductees. Using the given data, this calculates as \(47 / 303\).

Convert Fraction to Decimal

To express the proportion as a decimal, the fraction should be computed. Using a calculator, this evaluates as approximately 0.155. So, the proportion of inductees that have been female or have included female members is approximately 0.155 or 15.5%.

Expected Distribution of Sample Proportions

The distribution of sample proportions is expected to follow a normal distribution according to the Central Limit Theorem, given that the sample size is large enough, which in this case is 50. Since the population proportion is approximately 0.155, we expect the sample proportions to be centered around this value.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental statistical principle that plays a pivotal role in the field of statistics. It states that regardless of the population's distribution, as long as the sample size is sufficiently large, the distribution of the sample means will approach a normal distribution. This is essential when predicting the characteristics of populations based on sample data.

To bring this concept home for students using the 'Rock and Roll Hall of Fame' inductees example, imagine you're drawing repeated samples of 50 inductees. Even if the actual distribution of female inductees in the full population isn't normal, the distribution of the proportion of female inductees in each sample would tend toward a bell-shaped curve with enough samples. This theorem enables researchers to make inferences about population parameters, such as the mean and variability, which are crucial for decision-making processes in nearly every field of inquiry.

Normal Distribution

What exactly is the normal distribution? It's a probability distribution that is symmetrically shaped like a bell curve, where most of the observations cluster around the central peak, and the probabilities for values further away from the mean taper off equally in both directions. In statistics, the normal distribution is incredibly significant because of its predictability and the ease with which it can be worked with mathematically.

For instance, in our Rock and Roll Hall of Fame example, if we graphed the proportions of female or female-inclusive groups among a large number of sample sets of inductees, we'd expect the graph to take on that classic bell-shape. This is due to the characteristics of the normal distribution, where about 68% of values fall within one standard deviation of the mean, 95% within two, and almost all within three. Understanding this concept helps students grasp why we can use the sample to make reliable inferences about the entire population.

Sampling Distribution

A sampling distribution is not about the distribution of the population itself but the distribution of a statistic, like the mean or proportion, over many different samples taken from the population. Think of it as a collection of statistics, all pulled from their own unique samples, and aggregated into a new data set.

In the context of the textbook problem, if we assembled all possible samples of size 50 from the inductees of the Rock and Roll Hall of Fame and calculated the proportion of female members or female-inclusive groups in each sample, we're essentially creating a sampling distribution. This distribution would illustrate the variation of sample proportions, showing us how much those sample proportions can differ by chance alone. It's this concept that underpins the processes of hypothesis testing and the construction of confidence intervals, allowing statisticians to navigate the uncertainty inherent in working with samples.