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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. \(^{20}\) (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 population parameter

A sample statistic is a value that provides information about a sample, while a population parameter is a value that provides information about the population as a whole. Since the 61% value is based on responses from 1,578 US residents (a sample of the entire US population), it is considered a sample statistic, not a population parameter.

02

Calculate the standard error for the proportion

The formula for the standard error of a proportion is \(SE = \sqrt{\frac{p \cdot (1-p)}{n}}\), where \(p = 0.61\) and \(n = 1578\). Calculate this to get \(SE \approx \sqrt{\frac{0.61 \cdot 0.39}{1578}} \approx 0.0123\).

03

Construct the 95% confidence interval

A 95% confidence interval for the proportion is given by \(p \pm Z \times SE\), where \(Z = 1.96\) for 95% confidence level. This calculates to \(0.61 \pm 1.96 \times 0.0123\), resulting in a confidence interval of approximately \(0.586\) to \(0.634\).

04

Evaluate the assumption of normal distribution

For a normal approximation to be reasonable, both \(np\) and \(n(1-p)\) should be greater than 10. Here, \(np = 1578 \times 0.61 = 962.58\) and \(n(1-p) = 1578 \times 0.39 = 615.42\), both of which are greater than 10, thus satisfying the condition.

05

Analyze the news statement

The confidence interval ranging from 0.586 to 0.634 suggests that more than 50% of the population supports legalization, as the entire interval is above 0.5. Therefore, the statement "Majority of Americans think marijuana should be legalized" is justified based on this confidence interval.

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Key Concepts

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

Sample Statistics

In statistics, the term 'sample statistic' refers to a numerical value calculated from a subset of data, known as a sample. This subsample comes from a larger group called the population. In the context of the General Social Survey (GSS), where 1,578 US residents were surveyed on their views about marijuana legalization, the percentage of respondents who supported legalization (61%) represents a sample statistic.

This number was derived from a specific sample of individuals rather than all US residents.

• The sample statistic provides an estimate that helps in understanding or making inferences about the entire population.
• In practice, working with a sample is much more feasible than trying to measure a whole population.
• Because of this, researchers use sample statistics to make educated guesses about population parameters, but it's important to remember that these estimates are subject to sampling error.

Population Parameters

The concept of 'population parameters' involves understanding that they are values that describe certain characteristics of the whole population. Unlike sample statistics, population parameters represent the true value or behavior within the entire population.

However, often these are unknown because it's generally impractical to collect data from every member of the population. In our case with the GSS, the true proportion of all US residents who think marijuana should be legalized is the population parameter of interest.

Due to the impractical nature of polling every individual, researchers use data from samples like the survey of 1,578 residents to estimate these parameters.

• Population parameters are the 'target' values statistician aim to estimate using sample data.
• These parameters are fixed but unknown, whereas sample statistics are known but variable because they change with each sample.

Normal Distribution

Normal distribution is a core concept in statistics, often referred to as a bell curve. It describes how values of a variable are dispersed, conforming to a symmetrical, bell-shaped curve. In the context of the GSS survey, assuming a normal distribution helps provide a reliable estimation of the population proportion through confidence intervals.

For a sample proportion, the condition to approximate a normal distribution is that both the product of the sample size and the estimated proportion ( = 1578, p = 0.61) and its complement should exceed 10. Based on calculations, with values found to satisfy these conditions, the sample proportion can be modeled as normally distributed. This is crucial for calculating confidence intervals accurately.

• The normal distribution assumption allows the application of the Z-score in confidence interval calculations.
• It simplifies analysis by converting raw data into standard scores, facilitating interpretation.
• However, it's essential to verify that the sample data indeed fits the normal distribution criteria before proceeding with this method.

General Social Survey

The General Social Survey (GSS) is an important data source that provides insights into public opinion and societal trends across the United States. It has been collecting data from a representative sample of residents since 1972, covering a wide range of topics, including social, political, and economic issues.

In the case of marijuana legalization, the GSS asks people nationwide for their opinions. This survey is especially beneficial for deriving insights, as it represents a diverse cross-section of the country's demographics.

• Studies like the GSS help policymakers understand public sentiment on various issues.
• By using surveys to collect data from a large sample, GSS results can be generalized to reflect national trends.
• It’s important to communicate findings properly, considering both sample limitations and confidence intervals, as done in the marijuana example to ensure accurate and actionable conclusions.