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Chapter 3: Problem 40

A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.38 tells us that 206 of the 303 inductees to the Rock and Roll Hall of Fame have been performers. 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 performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of \(p=0.68 ?\) 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

Expert verified
Without the specific results from a statistical calculator or software, the short answer can't be definitively provided. However, it is expected that as the sample size increases, the standard error will decrease. This would mean that the accuracy of the sample proportion as an estimate of the population proportion improves (the sample proportions should get closer to \(p=0.68\)). Therefore, the size of the sample contributes significantly to the accuracy of the estimates.

Step by step solution

01

Sampling Distribution for \(n=10\)

Take many random samples from the population of size \(n=10\). For each sample, calculate the sample proportion that are performers and calculate the standard error. The standard error can be calculated using the formula \(SE = \sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size. Identify the value of the sample proportion farthest from the population proportion of \(p=0.68\).
02

Sampling Distribution for \(n=20\)

Repeat the same process as in Step 1, but this time with a sample size of \(n=20\). Calculate the sample proportion and standard error for each sample and identify the sample proportion that deviates the most from \(p=0.68\).
03

Sampling Distribution for \(n=50\)

Follow the same procedure as in the previous steps, but with a larger sample size of \(n=50\). Again, calculate the sample proportion and standard error for each sample and find the sample proportion that differs the most from \(p=0.68\).
04

Analyzing the Impact of Increasing Sample size on Accuracy

Using the results obtained in the previous steps, analyze how the sample size affects the accuracy of estimating the population proportion. Pay attention to how the standard error decreases as the sample size increases - a smaller standard error indicates a more accurate estimate of the population proportion. Also, note if the sample proportions that are the farthest from \(p=0.68\) get closer to the population proportion as the sample size increases. This would also indicate an increase in accuracy.

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Key Concepts

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

Standard Error
The standard error is a measure that tells us how much our sample statistic is expected to fluctuate from the true population parameter. When we take samples from a population, the intent is to estimate a statistic, such as a mean or a proportion. The standard error gives us an idea of the precision of our sample estimate.
For example, suppose we want to estimate the proportion of Rock and Roll Hall of Fame inductees who are performers. In mathematical terms, the standard error of sample proportions is given by the formula: \[SE = \sqrt{\frac{p(1 - p)}{n}}\]where:
  • \( p \) is the population proportion.
  • \( n \) is the sample size.

As the sample size \( n \) increases, the standard error decreases, indicating more precise estimates. This means that as you increase your sample size from 10 to 50, your estimates of the sample proportion will become more reliable and closer to the true population proportion.
Sample Proportion
The sample proportion is the fraction of individuals in a sample that possess a particular characteristic. In the context of the Rock and Roll Hall of Fame, if out of a sample of 10 inductees, 7 are performers, the sample proportion is 0.7 or 70%.
Calculating the sample proportion is straightforward. For each sample, count the individuals with the characteristic of interest and divide by the total number of individuals in the sample.
The sample proportion is critical because it serves as an estimate of the population proportion. However, remember that different samples can provide different sample proportions. This variability is where the standard error plays a crucial role as it helps quantify this variability.
Larger samples are generally more representative of the population, thus the sample proportions from these samples tend to be closer to the actual population proportion.
Population Proportion
The population proportion refers to the ratio of individuals in the entire population that have a specific characteristic. In our exercise, it is given that the population proportion \( p \) is 0.68, meaning that 68% of all inductees in the Rock and Roll Hall of Fame are performers.
Unlike sample proportions that may vary with each sample, the population proportion is a fixed value. It represents the true extent of the characteristic within the whole population and is often unknown. That's why sampling is used to make estimates.
Knowing the population proportion allows us to assess the accuracy of our sample estimates. For example, if your sample proportion is significantly different from \( p \), it might suggest that your sample is not representative, or it's just due to natural sampling variability.
By increasing the sample size, we enhance the matching of sample proportions to the population proportion, demonstrating the importance of larger samples for more accurate and reliable statistics.

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Most popular questions from this chapter

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