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Chapter 8: Problem 54

Population variability Explain the reasoning behind the following statement: "In studies about a very diverse population, large samples are often necessary, whereas for more homogeneous populations smaller samples are often adequate." Illustrate for the problem of estimating mean income for all medical doctors in the United States compared to estimating mean income for all entry-level employees at McDonald's restaurants in the United States.

Short Answer

Expert verified
Diverse populations require larger samples for accurate estimates due to variability, while homogeneous populations need smaller samples.

Step by step solution

01

Understanding Population Variability

The statement is about how diverse (variable) or homogeneous a population is. In a diverse population, there's a wide range of characteristics or values, making it more challenging to capture accurate estimates with smaller samples. Conversely, homogeneous populations have similar characteristics, making smaller samples more reliable.
02

Comparing Population Characteristics

Medical doctors in the US have varying specializations, experience levels, and work locations, leading to a wide range of incomes. In contrast, entry-level employees at McDonald's are likely to have more similar job roles, work experiences, and therefore similar incomes across locations.
03

Impact on Sample Size Requirements

When estimating the mean income of diverse populations (like medical doctors), larger samples are needed to account for the high variability and ensure the sample accurately reflects the entire population. For more homogeneous populations (like entry-level McDonald's employees), smaller samples may suffice because there's less income variability to capture.
04

Illustrating with Examples

To estimate the mean income of US medical doctors accurately, a large sample is needed to represent the variances in wages across specialties and locations. However, for McDonald's employees, a smaller sample could effectively estimate the income because of the uniformity in entry-level pay across the nation.

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Key Concepts

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

Sample Size
Sample size refers to the number of observations or data points collected for a study. It plays a crucial role in the reliability and validity of the results. A larger sample size tends to give more reliable results because it better represents the population, capturing more of its natural variability.
- Larger sample sizes tend to provide more accurate estimates and reduce the margin of error. - Smaller sample sizes may lead to greater uncertainty and less reliable results.
In studies where populations are diverse, slight differences can greatly affect the result, thus a larger sample size becomes necessary. Conversely, if the population is homogeneous, even small samples can accurately reflect the entire population due to the similarity of characteristics among the members.
Homogeneous Population
A homogeneous population is one where members have similar characteristics or traits. This means there's less diversity in the variable you are studying.
For example, if you consider the population of entry-level employees at McDonald's, their incomes are quite similar. This results from standardized pay scales and uniform job roles.
- Since most of the members share similar attributes in a homogeneous population, smaller samples are generally sufficient to make accurate estimates. - Homogeneity reduces the variability within the data, making it easier to predict.
With such populations, researchers can confidently draw conclusions from smaller sample sizes, as the lack of variability means the sample is more likely to be representative of the whole population.
Diverse Population
In a diverse population, there is a wide range of different characteristics or values among its members. High variability can make it challenging to understand overall trends without a larger sample size.
Take, for example, medical doctors in the United States. They can vary greatly in terms of specialization, experience, and geographic location, leading to a wide range of incomes.
- A diverse population requires a larger sample size to ensure all types of members are included. - Larger samples help capture the wide variability and provide more accurate estimates.
When studying such populations, it's crucial to gather enough data to account for different variables that can significantly change the outcome or understanding of the data, ensuring that every nuance is captured in the study.

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Most popular questions from this chapter

According to a survey conducted by KPMG in \(2016,\) almost \(46 \%\) of the surveyed companies intend to grow their operations in Luxembourg over the next two years (Source: https://www.kpmg.com/LU/en/IssuesAndInsights/ Articlespublications/Documents/ManagementCompany-CEO-Survey-032016.pdf). a. Can you specify the assumptions made to construct a \(95 \%\) confidence interval for the population proportion? b. If the sample size is 5000 , verify that the assumptions of part a are satisfied and construct the \(95 \%\) confidence interval. Determine whether the proportion of companies who intend to grow their operations in Luxembourg is a majority or a minority.

A national survey was conducted by the Pew Research Center (www.people-press. org) between February \(18-21,2016 .\) Among 1002 participating adults, \(51 \%\) said that Apple Inc. should assist the FBI in their investigations by unlocking the iPhone used by one of the suspects in the San Bernardino terrorist attacks. Based on these data, can we conclude that more than half of Americans support the Department of Justice over Apple Inc. in this dispute over unlocking the concerned iPhone? Explain.

A Gallup poll of 2000 Russians taken between April and June 2014 (after the Olympic games in Russia and the annexation of the Crimean peninsula) showed that \(83 \%\) approved of President Putin's performance. If random sampling were used for this survey, what is the margin of error for this estimate at the \(95 \%\) confidence level? (In fact, Gallup uses weighted sampling and, based on it, reports a margin of error of \(2.7 \% .\) ) Source: gallup. com and search for "Russian Approval."

Using the Explore Coverage web app, let's check that the large-sample confidence interval for a proportion may work poorly with small samples. In the app, set \(p=0.30\), \(n=10\) and leave the confidence level at \(95 \% .\) Select to draw 100 random samples of size \(n\) and then click on Draw Sample(s). a. How many of the intervals you generated with the app fail to contain the true value, \(p=0.30 ?\) b. How many would you expect not to contain the true value? What does this suggest? c. To see that this is not a fluke, now take 1000 samples and see what percentage of \(95 \%\) confidence intervals contain \(0.30 .\) (Note: For every interval formed, the number of successes is smaller than \(15,\) so the large- sample formula is not adequate.) d. Using the Sampling Distribution for a Sample Proportion web app, generate 10,000 random samples of size 10 when \(p=0.30\). The app will plot the simulated sampling distribution of the sample proportion values. Is it bell shaped? Use this to help you explain why the large-sample confidence interval performs poorly in this case. (This exercise illustrates that assumptions for statistical methods are important, because the methods may perform poorly if we use them when the assumptions are violated.)

Google Glass Google started selling Google Glass (a type of wearable technology that projects information onto the eye) in the United States in spring 2014 for \(\$ 1500\). A national poll reveals that out of 500 people sampled, nobody owned Google Glass. a. Find the sample proportion who don't own Google Glass and its standard error. b. Find a \(95 \%\) confidence interval, using the large-sample formula. Is it sensible to conclude that no one in the United States owns Google Glass?] c. Why is it not appropriate to use the ordinary large-sample confidence interval in part b? Use a more appropriate approach and interpret the result. d. Is it plausible to say that fewer than \(1 \%\) of the population own Google Glass? Explain.
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