91Ó°ÊÓ

Life rating in Greece. Greece has faced a severe economic crisis since the end of \(2009 .\) A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that \(25 \%\) of them said they would rate their lives poorly enough to be considered "suffering". (a) Describe the population parameter of interest. What is the value of the point estimate of this parameter? (b) Check if the conditions required for constructing a confidence interval based on these data are met. (c) Construct a \(95 \%\) confidence interval for the proportion of Greeks who are "suffering". (d) Without doing any calculations, describe what would happen to the confidence interval if we decided to use a higher confidence level. (e) Without doing any calculations, describe what would happen to the confidence interval if we used a larger sample.

Step by step solution

01

Identify the Population Parameter

The population parameter of interest is the true proportion of Greeks in 2011 who rate their lives poorly enough to be considered 'suffering'. This is what we aim to estimate using the sample data.

02

Point Estimate

The point estimate for the population proportion, based on the sample, is the sample proportion. With 1,000 Greeks surveyed, 25% said they were 'suffering'. Hence, the point estimate is \(\hat{p} = 0.25\).

03

Check Conditions for Confidence Interval

The conditions for constructing a confidence interval for a proportion are: 1. Random Sample: The sample of 1,000 Greeks is a random sample.2. Large Enough Sample: The sample size is large enough if \(np \geq 10\) and \(n(1-p) \geq 10\). Here, \(np = 1,000 \times 0.25 = 250\) and \(n(1-p) = 1,000 \times 0.75 = 750\), both greater than 10.Both conditions are satisfied, so we can proceed.

04

Construct Confidence Interval

To construct a 95% confidence interval for the proportion:1. Find the standard error: \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.25 \times 0.75}{1,000}} = 0.0137\).2. Find the critical value for a 95% confidence level (\(z\)-score ≈ 1.96).3. Compute the margin of error: \(ME = 1.96 \times 0.0137 = 0.0268\).4. The confidence interval is: \(\hat{p} \pm ME = 0.25 \pm 0.0268 = (0.2232, 0.2768)\).

05

Effect of Higher Confidence Level

Increasing the confidence level would require a larger critical value, which increases the margin of error. Therefore, the confidence interval would be wider, indicating more uncertainty in the estimate.

06

Effect of Larger Sample

Using a larger sample reduces the standard error because \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). This leads to a smaller margin of error, thus resulting in a narrower confidence interval, providing a more precise estimate of the population proportion.

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Key Concepts

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

Population Parameter

In the context of statistical analysis, a population parameter is a value that represents a characteristic of an entire population. It's often unknown and is what researchers aim to estimate based on sample data. Here, the population parameter of interest is the true proportion of Greeks in 2011 who rate their lives as 'suffering' during the economic crisis.

This parameter gives us a picture of the overall well-being of Greeks at the time and guides social scientists to understand and address issues related to the crisis. Estimating this parameter accurately helps in making data-informed decisions and understanding the societal impact of economic conditions.

Point Estimate

A point estimate provides a single value estimate of a population parameter. It is calculated from sample data and serves as the best guess of the true population parameter. In the given scenario, the point estimate is derived from the Gallup poll where 1,000 Greeks were surveyed, and 25% reported 'suffering'.

The calculation is straightforward: the point estimate \( \hat{p} = 0.25 \). This is based on the sample proportion, meaning that approximately 25% of the sampled Greeks considered themselves to be suffering.

• This point estimate is used as a basis to construct confidence intervals, offering insights with a quantifiable likelihood.

It represents the center of a confidence interval, which provides a range of values for better estimating the true population parameter.

Gallup Poll

The Gallup Poll is a reputable survey-based research method commonly used to assess public opinion on various social issues, including economic conditions, health, and quality of life. It helps to gauge the sentiment of a population through random samples and rigorous methods.

• Gallup ensures a representative sample by randomly selecting individuals, which increases the validity of the survey results.
• Results provide a snapshot of how certain situations, like the economic crisis in Greece, impact citizens.

In this exercise, the Gallup Poll was central to gathering data from Greek citizens, allowing researchers to estimate the level of suffering during a specific period. Such polls are crucial for providing policy-makers with the necessary insights to craft informed strategies and interventions aimed at alleviating such societal challenges.

Standard Error

The standard error (SE) is a statistic that measures how much the sample proportion is expected to fluctuate from the true population proportion. It's a key component in determining how precise our point estimate is. The smaller the standard error, the more reliable the point estimate becomes.

For this Gallup poll scenario, the formula used is \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. Here, the SE was calculated to be approximately 0.0137, indicating how much the sample proportion might differ from the actual population proportion.

• A lower SE suggests less variability in our estimates, leading to narrower confidence intervals, thus providing more precise estimates about the population.
• If we had a larger sample size, the SE would decrease, further enhancing the accuracy of our predictions.

Understanding and calculating SE is essential for constructing confidence intervals and assessing the accuracy of estimates.