Question

A Sampling Distribution for Gender in the Rock and Roll Hall of Fame Exercise 3.37 tells us that 47 of the 303 inductees to the Rock and Roll Hall of Fame have been female or have included female members. The data are given in RockandRoll. Using all inductees as your population: (a) Use StatKey or other technology to take many random samples of size \(n=10\) and compute the sample proportion that are female or with female members. What is the standard error for these sample proportions? What is the value of the sample proportion farthest from the population proportion of \(p=0.155 ?\) How far away is it? (b) Repeat part (a) using samples of size \(n=20\). (c) Repeat part (a) using samples of size \(n=50\). (d) Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

Short answer

Since the exact solution will vary on the basis of random sampling, a general answer would be that increasing the sample size provides a better, more accurate estimate of the population proportion. As sample size increases, the standard error decreases, which demonstrates that the sample proportion will be closer to the population proportion. Additionally, increasing the sample size decreases the chance of getting a sample far from the population proportion.

Step-by-step solution

Analyze and establish the population proportion

The first step is to understand the population sample. From 303 inductees, 47 are female or include female members. Therefore, the population proportion of female inductees (\(p\)) can be computed as \(p = \dfrac{47}{303} = 0.155\).

Calculate the sampling distribution for \(n=10\)

Here, we have to use technology like StatKey or similar, to take many random samples of size \(n=10\) from the given population. We then compute the sample proportion (\(\hat{p}\)) for each sample. Moreover, we are supposed to find out the standard error for these samples, which we will calculate as the standard deviation of the sample proportions. Find out the value of the sample proportion (\(\hat{p}\)) that is furthest from the population proportion \(p = 0.155\) and note down how far away it is from \(p\).

Calculate the sampling distribution for \(n=20\)

Repeat step 2, but this time with samples of size \(n=20\). Note the changes in the values of \(\hat{p}\), the standard error, and the distance of the farthest \(\hat{p}\) from \(p\).

Calculate the sampling distribution for \(n=50\)

Repeat step 2 once again, but now with samples of size \(n=50\). Again, observe the effect of a larger sample size on the values of \(\hat{p}\), the standard error, and the distance of the farthest \(\hat{p}\) from \(p\).

Compare the values for different sample sizes

At this point, we have calculated and accumulated a lot of data. Analyze and compare the results from each different sample size and evaluate the effect of increasing the sample size on the accuracy of using the sample proportion to estimate the population proportion.

Key concepts

Population Proportion

When studying statistics, one common task is estimating the characteristics of a large group, or population, by examining a smaller group, or sample. The population proportion is a measure that represents the proportion of a particular characteristic in a population. For example, in the Rock and Roll Hall of Fame exercise, we were given that out of 303 inductees, 47 are female or include female members, which gives us a population proportion (\( p \)) of 0.155 for female inductees.

Understanding the population proportion is essential because it serves as a benchmark for comparing the results of samples. When a sample is drawn and analyzed, we can calculate the sample proportion, denoted as \( \hat{p} \), to see how closely it aligns with the population proportion. This comparison is fundamental in determining how well a sample represents the population.

Sampling Error

One of the inherent aspects of drawing samples is that they won't perfectly represent the population due to chance variation. This difference is known as the sampling error. Technically, sampling error is defined as the difference between the sample statistic and the actual population parameter it is intended to estimate.

In our exercise about the gender distribution in the Rock and Roll Hall of Fame, when we draw multiple samples and calculate the sample proportion \( \hat{p} \), not all of them will be equal to the population proportion \( p = 0.155 \). Some sample proportions will be higher, and some will be lower. The standard error gives us a way to quantify this variability; it is essentially the standard deviation of the sample proportions. As we increase the size of the sample, the standard error generally decreases, which means the sample proportions are likely to be closer to the population proportion, providing a higher accuracy in our estimation.

Sample Size Effect

The size of the sample has a profound effect on the quality of the statistical estimates that can be made about the population. This phenomenon is known as the sample size effect. Larger samples tend to provide more accurate and reliable estimates of the population parameters, simply because they contain more data points from the population.

As seen in the exercise, by increasing the sample size from \( n=10 \) to \( n=20 \), and then to \( n=50 \), we observe a trend: the standard error of the sample proportion decreases. This implies that the distribution of \( \hat{p} \) for larger samples will be more tightly clustered around the population proportion \( p = 0.155 \), indicating a lower sampling error. Consequently, larger samples lead to estimates that are more likely to be close to the true population proportion, thus making them more reliable for making inferences about the population.