91Ó°ÊÓ

A Sampling Distribution for Gender in the Rock and Roll Hall of Fame Exercise 3.35 tells us that 41 of the 273 inductees to the Rock and Roll Hall of Fame have been female or have included female members. The data are given in Rockand Roll. 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 \(p=0.150 ?\) 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.

Step by step solution

01

Calculation of the Population Proportion

Calculate the population proportion \(p\) first. It’s found by dividing the number of successful outcomes (inductees that are female or with female members) by the total number of outcomes. Hence, \(p = 41/273\).

02

Sampling and Calculating Sample Proportions

Take several random samples each of size \(n=10\), \(n=20\), and \(n=50\) from the population and calculate the sample proportions for each sample that are female or with female members. Use technological tools for the random sampling process and repeat the process many times for each sample size.

03

Calculation of Standard Error and Distance

Calculate the standard error for the sample proportions for each sample size. Remember, the standard error of a proportion is given by \(\sqrt{p(1-p)/n}\). Also, figure out the sample proportion for each size that’s farthest from the population proportion \(p\), and quantify how far away it is.

04

Comparing the Effects of Sample Size

Repeat Steps 2 and 3 for sample sizes \(n=20\) and \(n=50\) just like for \(n=10\). This helps to compare the effects of these different sample sizes on sampling distribution of the proportions and hence, the accuracy of the estimates.

05

Observation and Conclusion

Observe and draw conclusions about how the standard error and the distance between the farthest sample proportion and the population proportion change as the sample size increases. From this, we can understand how accuracy of estimation changes with sample size.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Proportion

Understanding the term population proportion is crucial in statistics. It refers to the proportion of members in a total population who have a certain characteristic. For example, if we are analyzing the Rock and Roll Hall of Fame inductees, as in the original exercise, and we want to know the proportion of inductees that are female or include female members, we calculate it by dividing the number of those specific inductees by the total number.

If there are 41 inductees that are female or include female members out of 273 total inductees, the population proportion (\(p\)) is \(p = \frac{41}{273}\). This ratio provides a benchmark for analyzing sample proportions and understanding how well they represent the overall population.

Sample Proportion

The sample proportion is similar to the population proportion but is calculated for a subset of the population, known as a sample. If we take a sample of the Rock and Roll Hall of Fame inductees, for instance, 10 inductees at random, and determine how many are female or with female members, that would give us the sample proportion.

Random samples can yield varying proportions which then create a sampling distribution when we consider many samples. Understanding the sample proportion, especially in relation to the population proportion, helps in assessing the representativeness of the sample and the reliability of predictions made about the wider population.

Standard Error

The concept of standard error is pivotal in statistics, as it measures the deviation of a sample statistic (like the sample proportion) from the population parameter it estimates (like the population proportion). In the context of our exercise with the Rock and Roll Hall of Fame, standard error lets us quantitatively grasp the variability of sample proportions from the actual population proportion.

Using the formula for the standard error of the proportion, \( SE = \sqrt{\frac{p(1-p)}{n}} \), where \(p\) is the population proportion and \(n\) is the sample size, we can calculate how much variability to expect. As sample size increases, the standard error usually decreases, signaling more precise estimations.

Sample Size

The sample size, denoted as \(n\), plays a decisive role in statistical accuracy. It refers to the number of observations included in a sample drawn from a population. From the exercise, when investigating the proportion of female inductees in the Rock and Roll Hall of Fame, we consider samples of 10, 20, and 50 individuals.

The size of the sample affects the variability of the sample proportion, with larger samples tending to yield more stable and accurate estimates. This is because larger samples are more likely to be representative of the population, leading to smaller standard errors and more reliable conclusions.

Random Sampling

Random sampling is a technique whereby each member of a population has an equal chance of being selected in the sample. This method is fundamental to obtain unbiased results. In the Rock and Roll Hall of Fame exercise, random sampling helps ensure that each sample proportion of female or with female members reflects the diversity and variability present in the full population.

By taking many random samples of different sizes and calculating their proportions that are female, we construct a clear picture of the sampling distribution. This helps in understanding how sample proportions may fluctuate around the population proportion simply by chance, informing the reliability of our estimates.