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Chapter 8: Problem 6

A random sample of 100 employees of a large company included 37 who had worked for the company for less than one year. For this sample, \(\hat{p}=\frac{37}{100}=0.37 .\) If a different random sample of 100 employees were selected, would you expect that \(\hat{p}\) for that sample would also be \(0.37 ?\) Explain why or why not.

Short Answer

Expert verified
In conclusion, we cannot expect that the proportion (\(\hat{p}\)) for a different random sample of 100 employees would be exactly 0.37, due to the inherent sampling variability in random sampling. However, the proportions would likely be close to each other if the sample size is large enough and the sampling is done randomly.

Step by step solution

01

In this problem, we have a random sample of 100 employees: - Total employees in the sample (n) = 100 - Employees who worked for less than one year (x) = 37 - Proportion of employees who worked for less than one year (\(\hat{p}\)) = \(\frac{37}{100}\) = 0.37 Now, we need to find out if a different random sample of 100 employees would also have this same proportion of 0.37. #Step 2: Understanding the concept of sampling variability#

When taking random samples from a population, it is important to understand that the sample statistics (like \(\hat{p}\)) can vary from one sample to another simply due to the randomness in the sampling process. This variability is known as sampling variability. The sampling distribution of a sample statistic provides the information about the variability of the statistic across different samples. In our case, it would be the distribution of \(\hat{p}\) across different samples of 100 employees. #Step 3: Discussing the variability in \(\hat{p}\) for different samples#
02

It is unlikely that a different random sample of 100 employees would have exactly the same proportion (\(\hat{p}\)) of 0.37 for employees who have worked for the company for less than one year. This is because of the sampling variability, as different random samples from the population can yield slightly different results in the statistics due to randomness. Although the proportions might not be exactly equal, they would be close to each other if the sample size is large enough and the sampling is done randomly. We can expect that the distribution of the sample proportions will tend to be centered around the true population proportion, and the variability will be smaller when the sample size is larger. #Step 4: Conclusion#

In conclusion, we cannot expect that the proportion (\(\hat{p}\)) for a different random sample of 100 employees would be exactly 0.37. This is due to the inherent sampling variability in random sampling. However, the proportions would likely be close to each other if the sample size is large enough and the sampling is done randomly.

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Key Concepts

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

Random Sampling
Random sampling is like picking names from a hat. It ensures that each member of a population has an equal chance of being selected. This method helps to avoid bias and ensures that the sample represents the entire population.

For example, in our exercise, we picked 100 employees randomly from a large company. This means that these 100 employees should reflect the characteristics of all employees in the company.

Random sampling is crucial because it allows for generalizations about a population based on the sample data. However, due to randomness, there's always a bit of uncertainty in the results.
Sample Proportion
The sample proportion is a statistic that describes a part of the whole sample. It's denoted by \( \hat{p} \) and represents the ratio of a particular group within your sample.

In our example, \( \hat{p} = 0.37 \) indicates that 37% of the sampled employees have worked for less than one year. We calculate it by dividing the number of employees who fit the criteria by the total number of employees in the sample.

Sample proportions help us understand the parts of a population when only the sample data is available. Remember, \( \hat{p} \) will fluctuate with different samples due to sampling variability.
Sampling Distribution
The sampling distribution of a sample statistic is like a map of possible outcomes. It shows the distribution of the sample proportion (\( \hat{p} \)) over many repetitions of the sample.

This means if we took many samples of 100 employees, each sample would provide a different \( \hat{p} \), and the collection of these \( \hat{p} \) values forms the sampling distribution.

Sampling distributions help us understand how much variability to expect from \( \hat{p} \). Generally, for large sample sizes, this distribution will be approximately normal, centered around the true population proportion.
Population Proportion
The population proportion is like the true answer to a question posed about an entire population. Represented by \( p \), it tells us the fraction of the whole population that exhibits a certain characteristic.

Unlike the sample proportion, the population proportion is constant for a given population. However, we rarely know its precise value, so we use the sample proportion to estimate it.

In our exercise, the true population proportion of employees with less than one year at the company could be different from 0.37. As we take more samples, we get closer to understanding this true value.

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Most popular questions from this chapter

Explain why the standard deviation of \(\hat{p}\) is equal to 0 when the population proportion is equal to 1 .

The report "A Crisis in Civic Education" (January 2016, goacta.org/images/download/A_Crisis_in_Civic_Education .pdf, retrieved May 3,2017 ) indicated that in a survey of a random sample of 1000 recent college graduates, 96 indicated that they believed that Judith Sheindlin (also known on TV as "Judge Judy") was a member of the U.S. Supreme Court. Is it reasonable to conclude that the proportion of recent college graduates who have this incorrect belief is greater than 0.09 \((9 \%) ?\) (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .)\)

The U.S. Census Bureau reported that in 2015 the proportion of adult Americans age 25 and older who have a bachelor's degree or higher is 0.325 ("Educational Attainment in the United States: 2015," www.census.gov, retrieved January 22,2017 ). Consider the population of all adult Americans age 25 and over in 2015 and define \(\hat{p}\) to the proportion of people in a random sample from this population who have a bachelor's degree or higher. a. Would \(\hat{p}\) based on a random sample of only 10 people from this population have a sampling distribution that is approximately normal? Explain why or why not. b. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is \(400 ?\) c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(400 .\) Does the change in sample size affect the mean and standard deviation of the sampling distribution of \(\hat{p} ?\) If so, what are the new values for the mean and standard deviation? If not, explain why not.

Consider the following statement: A county tax assessor reported that the proportion of property owners who paid 2016 property taxes on time was \(\mathbf{0} . \mathbf{9 3}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.93\) or \(\hat{p}=0.93 ?\)

In a study of pet owners, it was reported that 24\% celebrate their pet's birthday (Pet Statistics, Bissell Homecare, Inc., 2010). Suppose that this estimate was based on a random sample of 200 pet owners. Is it reasonable to conclude that the proportion of all pet owners who celebrate their pet's birthday is less than \(0.25 ?\) Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.
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