91Ó°ÊÓ

The General Social Survey asked 1,578 US residents: "Do you think the use of marijuana should be made legal, or not?" \(61 \%\) of the respondents said it should be made legal. 13 (a) Is \(61 \%\) a sample statistic or a population parameter? Explain. (b) Construct a \(95 \%\) confidence interval for the proportion of US residents who think marijuana should be made legal, and interpret it in the context of the data. (c) A critic points out that this \(95 \%\) confidence interval is only accurate if the statistic follows a normal distribution, or if the normal model is a good approximation. Is this true for these data? Explain. (d) A news piece on this survey's findings states, "Majority of Americans think marijuana should be legalized." Based on your confidence interval, is this news piece's statement justified?

Step by step solution

01

Determine if 61% is a sample statistic or a population parameter

A population parameter is a characteristic or measure obtained by using all of the data from a population, while a sample statistic is a characteristic or measure obtained by using the data from a sample. Since 61% refers to the proportion of the 1,578 sampled respondents and not the entire population, it is a sample statistic.

02

Calculate the 95% Confidence Interval

To construct a 95% confidence interval for the proportion of US residents who support marijuana legalization, use the formula for a confidence interval of a proportion: \(\hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)where \(\hat{p} = 0.61\), \(n = 1578\), and \(Z\) for a 95% confidence level is approximately 1.96.- Calculate the standard error: \(\sqrt{\frac{0.61 \cdot (1-0.61)}{1578}} \approx 0.0123\)- Calculate the margin of error: \(1.96 \cdot 0.0123 \approx 0.0241\)- The 95% confidence interval is: \(0.61 \pm 0.0241 = [0.5859, 0.6341]\) Thus, the confidence interval is [0.5859, 0.6341].

03

Check if the normal model is appropriate

The normal model is appropriate if both \(np\) and \(n(1-p)\) are greater than 5. For this survey:- Calculate \(np = 1578 \cdot 0.61 \approx 962.58\)- Calculate \(n(1-p) = 1578 \cdot 0.39 \approx 615.42\)Since both values are greater than 5, the normal approximation is appropriate for these data.

04

Evaluate the news statement

The confidence interval [0.5859, 0.6341] does not contain 0.5, meaning the interval suggests more than 50% of the population supports legalization. Thus, stating that "a majority of Americans think marijuana should be legalized" is justified by the confidence interval.

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Key Concepts

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

Sample Statistic

In the world of statistics, it's crucial to distinguish between sample statistics and population parameters. These terms might sound similar, but they represent different aspects of data analysis. A **sample statistic** is a measure that describes some characteristic obtained from a subset, or sample, of a population. In this case, when the survey reported that 61% of the 1,578 participants support marijuana legalization, this percentage is our sample statistic. It's derived strictly from those surveyed, not the entire population. Therefore, it provides an estimate or snapshot based on just a segment of the whole, rather than reflecting everyone’s views.

Population Parameter

The concept of a **population parameter** is essentially the counterpart to a sample statistic. While the sample statistic represents a portion of a group, the population parameter represents a characteristic of the entire population being studied. If we wanted a true population parameter in this scenario, we would need data from every US resident. However, such comprehensive data is often impractical to collect. Instead, we rely on statistical methods, like confidence intervals, to estimate these parameters. This distinction is foundational, helping us understand whether data reflects a sample or seeks to estimate broader population tendencies.

Normal Distribution

The **normal distribution** is a vital concept in statistics, often known as the bell curve due to its characteristic shape. The idea is that for many kinds of data, values will tend to cluster around the mean - the average value. In the context of the survey on marijuana legalization, the normal distribution concept helps in constructing a confidence interval. This requires the data to follow a pattern that can be approximated as normal, especially when sample sizes are large. A great rule of thumb is ensuring that both the expected number of 'successes' (people supporting legalization) and 'failures' (those opposing) are more than 5. As calculated:

• The number of supporters: \(np = 1578 \times 0.61 \approx 962.58\)
• The number of non-supporters: \(n(1-p) = 1578 \times 0.39 \approx 615.42\)

Both conditions are satisfied, confirming the data follows a normal distribution pattern.

Proportion Estimation

**Proportion estimation** is at the heart of understanding survey data like this. The 61% is our sample proportion, representing those in favor of legalization. Using this, we can estimate the proportion of the entire population that shares this view, giving us an insight into societal trends. We create a confidence interval to decide how confident we can be in our estimate. In this exercise, the computed confidence interval \([0.5859, 0.6341]\) allows us to say, with 95% confidence, that the true proportion of the population that supports legalization lies within this range. This calculated interval reassures us that a majority supports legalization, as it firmly exceeds 50% and provides a robust basis for making broader assertions.