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When the 2000 GSS asked subjects (variable GRNSOL) if they would be willing to accept cuts in their standard of living to protect the environment, 344 of 1170 subjects said yes. a. Estimate the population proportion who would answer yes. b. Find the margin of error for a \(95 \%\) confidence interval for this estimate. c. Find a \(95 \%\) confidence interval for that proportion. What do the numbers in this interval represent? d. State and check the assumptions needed for the interval in part \(c\) to be valid.

Short Answer

Expert verified
a. The estimated proportion is 0.294. b. Margin of error is 0.025. c. Confidence interval is (0.269, 0.319). d. Sample size is sufficient, conditions are met.

Step by step solution

01

Estimating Population Proportion

To estimate the population proportion of subjects who would answer "yes," use the formula for the sample proportion:\[p = \frac{x}{n}\]where \(x\) is the number of subjects who said "yes" (344), and \(n\) is the total number of subjects (1170). Calculate:\[p = \frac{344}{1170} \approx 0.294\]Thus, the estimated population proportion is approximately 0.294.
02

Finding Margin of Error

To find the margin of error (ME) for the 95% confidence interval, use the formula:\[ME = z^* \times \sqrt{\frac{p(1-p)}{n}}\]For a 95% confidence level, \(z^*\) is approximately 1.96. Use the estimated proportion \(p = 0.294\) and \(n = 1170\).Substitute the values into the formula:\[ME = 1.96 \times \sqrt{\frac{0.294 \times (1-0.294)}{1170}} \approx 0.025\]
03

Finding Confidence Interval

The 95% confidence interval is calculated as:\[CI = p \pm ME\]Substitute the estimated proportion and margin of error:\[CI = 0.294 \pm 0.025\]This gives an interval of approximately (0.269, 0.319). The numbers in this interval estimate the range in which the true population proportion likely falls with 95% confidence.
04

Checking Assumptions

For the confidence interval to be valid, the sampling distribution of the sample proportion needs to be approximately normal. The conditions are:1. The sample size \(n\) should be large enough. A rule of thumb is \(np \geq 10\) and \(n(1-p) \geq 10\).2. The data should be collected using random sampling.For this data:\(344 \geq 10\) and \(826 \geq 10\).These conditions are satisfied, indicating that the assumptions are met for the interval to be valid.

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Key Concepts

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

Population Proportion
The idea of a population proportion is fundamental in statistics. It is a way of estimating the fraction or percentage of a population that has a particular characteristic. In our exercise, we're interested in the proportion of people who are willing to accept cuts in their standard of living to protect the environment.
To estimate this population proportion, we look at the sample proportion, which is calculated using the formula:
  • \( p = \frac{x}{n} \)
  • Here, \( x \) is the number of subjects in the sample who said "yes" (344), and \( n \) is the total number of subjects surveyed (1170).
  • Substituting the numbers: \( p = \frac{344}{1170} \approx 0.294 \)
The estimated population proportion is thus approximately 0.294. This means that, based on our sample, we estimate that about 29.4% of the overall population would say "yes."
Margin of Error
The margin of error (ME) is a measure that shows the range of values we expect the true population proportion to fall within, based on the sample data. It reflects the precision of our estimate.
For calculating the margin of error for a confidence interval, we use this formula:
  • \( ME = z^* \times \sqrt{\frac{p(1-p)}{n}} \)
  • In our scenario, \( z^* \) for a 95% confidence interval is 1.96.
  • With our sample proportion \( p = 0.294 \) and \( n = 1170 \), we calculate:
  • \( ME = 1.96 \times \sqrt{\frac{0.294 \times (1-0.294)}{1170}} \approx 0.025 \)
This shows that our estimated proportion is quite precise, with a small margin of error of 0.025 (or 2.5%). This means we're fairly confident that the true population proportion is within 2.5% higher or lower than our estimate.
Sampling Distribution
In statistics, a sampling distribution describes how a sample statistic—like the sample proportion—varies when taken over different samples from the same population.
This concept helps us understand the behavior of sample estimates and conditions under which they are valid.
Our confidence in the sampling distribution being approximately normal is drawn from the Central Limit Theorem. For a sample proportion, this assumes:
  • The sample size is large enough, often checked with the condition \( np \geq 10 \) and \( n(1-p) \geq 10 \).
  • In our case: \( 344 \geq 10 \) and \( 826 \geq 10 \), meeting the condition.
Therefore, our sampling distribution can be approximated as normal, supporting our confidence interval calculations.
Random Sampling
Random sampling is a technique where each member of a population has an equal chance of being included in the sample. This method is key in ensuring that the sample is representative of the population.
It helps eliminate selection bias and makes the results of a sample study generalize well to the whole population.
In our case, assuming that data was collected using random sampling means:
  • We'll be confident in saying that our sample proportion truly reflects the population proportion.
  • The reliability of statistical results, like our confidence interval, is dependent on how well the random sampling method was executed.
Using random sampling ensures that our conclusions drawn from the sample will likely hold true for the population at large.

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