91Ó°ÊÓ

In 2018 Gallup reported that \(52 \%\) of Americans are dissatisfied with the quality of the environment in the United States. This was based on a \(95 \%\) confidence interval with a margin of error of 4 percentage points. Assume the conditions for constructing the confidence interval are met. a. Report and interpret the confidence interval for the population proportion that are dissatisfied with the quality of the environment in the United States in 2018 . b. If the sample size were larger and the sample proportion stayed the same, would the resulting interval be wider or narrower than the one obtained in part a? c. If the confidence level were \(90 \%\) rather than \(95 \%\) and the sample proportion stayed the same, would the interval be wider or narrower than the one obtained in part a? d. In 2018 the population of the United States was roughly 327 million. If the population had been half that size, would this have changed any of the confidence intervals constructed in this problem? In other words, if the conditions for constructing a confidence interval are met, does the population size have any effect on the width of the interval?

Step by step solution

01

Determine the confidence interval

To determine the confidence interval for the dissatisfaction rate, you need to subtract and add the margin of error from the rate. The margin of error is given as 4% points, and the dissatisfied rate is given as 52%. Therefore, the lower and upper bounds of the confidence interval are \(52% - 4% = 48%\) and \(52% + 4% = 56%\). Thus, the confidence interval is \(48%, 56% \). As this procedure was performed at a 95% confidence level, it means that we are 95% confident that the true population proportion of Americans who are dissatisfied with the quality of the environment in the United States in 2018 is between 48% and 56%.

02

Impact of sample size on the confidence interval

If the sample size were larger but the sample proportion remained the same, the confidence interval would be narrower. This is because as the sample size increases, the standard error decreases, which means the margin of error decreases, thus making the confidence interval narrower.

03

Impact of confidence level on the confidence interval

If the confidence level was 90% rather than 95%, but the sample proportion remained the same, the confidence interval would be narrower. Lowering the confidence level leads to a smaller z-score, which in turn decreases the margin of error, which makes the confidence interval more narrow.

04

Impact of the population size on the confidence interval

If the population size were smaller, this would not change the width of the confidence interval, assuming that the conditions for constructing a confidence interval are met. The size of the population does not affect the width of the confidence interval, provided that the sample is far less than 10% of the population, which is very likely given the population was about 327 million in 2018.

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Key Concepts

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

Population Proportion

When discussing confidence intervals, the population proportion is an essential concept. In the exercise, the population proportion represents the percentage of Americans dissatisfied with the environmental quality in 2018, which is reported as 52%. This is based on the data collected from a sample of that population. It is important to understand that this proportion is only an estimate of the true proportion in the entire population.

The true population proportion is unknown, but we gauge it through our sample data. By using confidence intervals, we attempt to capture this true proportion within a certain range. For instance, in the example provided, the confidence interval was calculated to begin from 48% and end at 56%. This implies that we are reasonably certain, given the data, that the actual dissatisfaction rate falls somewhere within this range.

• The population proportion is a point estimate derived from sample data.
• Confidence intervals help provide a range where the true proportion is expected to lie.
• Accurate sample representation is crucial for determining reliable population proportions.

Margin of Error

The margin of error plays a pivotal role in the calculation of a confidence interval. In this exercise, the margin of error was determined to be 4 percentage points. It is the value you add and subtract from the sample proportion to construct the confidence interval.

The margin of error accounts for the degree of uncertainty or variability in the sample's estimate of the population proportion. It is influenced by the standard error, which itself is affected by factors like sample size and variability within the data. A smaller margin of error indicates a more precise estimate of the population proportion, whereas a larger margin implies less certainty.

• A greater margin of error means more variability and less confidence in our precision.
• Reducing the margin of error tightens the confidence interval, giving a more accurate portrayal of the population.
• The margin is a crucial measure of the reliability of our estimation.

Sample Size Impact

The size of the sample directly influences the confidence interval's width. In the exercise, it is acknowledged that if the sample size were larger, the interval would be narrower. This stems from the fact that a larger sample size reduces the standard error, subsequently decreasing the margin of error and resulting in a tighter interval.

Sample size impacts:

• A larger sample size gives a more accurate approximation of the population, leading to a narrower confidence interval.
• A smaller sample increases uncertainty and results in a wider interval.
• In effort to increase the precision of the confidence interval, increasing the sample size is often beneficial.

Remember that if the sample is significantly smaller, the estimates may be less reliable, potentially misrepresenting the true population characteristics. However, once the sample is reasonably large, increases in size have less impact on the interval's width.

Confidence Level Impact

The confidence level determines the degree of certainty we have that the confidence interval contains the true population proportion. In this exercise, a 95% confidence level was used. This high confidence level means we can be fairly certain that the true proportion is contained within the interval of 48% to 56%.

However, if the confidence level were decreased to 90%, the confidence interval would become narrower. This is because a lower confidence level corresponds to a smaller z-score, which reduces the margin of error.

• A higher confidence level offers more assurance but results in a wider interval.
• Lowering the confidence level narrows the interval, implying less certainty about the estimate.
• Choosing a confidence level entails balancing confidence with precision.

Selecting an appropriate confidence level is vital; it impacts the reliability and applicability of the confidence interval to real-world interpretations.