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Prop 19 in California. In a 2010 Survey USA poll, \(70 \%\) of the 119 respondents between the ages of 18 and 34 said they would vote in the 2010 general election for Prop \(19,\) which would change California law to legalize marijuana and allow it to be regulated and taxed. At a \(95 \%\) confidence level, this sample has an \(8 \%\) margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning." (a) We are \(95 \%\) confident that between \(62 \%\) and \(78 \%\) of the California voters in this sample support Prop \(19 .\) (b) We are \(95 \%\) confident that between \(62 \%\) and \(78 \%\) of all California voters between the ages of 18 and 34 support Prop \(19 .\) (c) If we considered many random samples of 119 California voters between the ages of 18 and 34 , and we calculated \(95 \%\) confidence intervals for each, \(95 \%\) of them will include the true population proportion of Californians who support Prop \(19 .\) (d) In order to decrease the margin of error to \(4 \%\), we would need to quadruple (multiply by 4) the sample size. (e) Based on this confidence interval, there is sufficient evidence to conclude that a majority of California voters between the ages of 18 and 34 support Prop \(19 .\)

Step by step solution

01

Understanding the Problem

We have a poll in which 119 respondents were surveyed, and 70% said they support Prop 19. The margin of error is 8% at a 95% confidence level.

02

Calculating the Confidence Interval

The confidence interval is calculated by taking the sample proportion and adding or subtracting the margin of error. Here, the sample proportion is 70% (or 0.70). The interval is thus \(0.70 \pm 0.08\), which means the interval is from 0.62 to 0.78, or 62% to 78%.

03

Analyzing Statement (a)

Statement (a) claims the confidence interval applies to the sample. Since confidence intervals are used to infer about the population, not the sample, this statement is false.

04

Analyzing Statement (b)

Statement (b) claims about all voters in the target population. The confidence interval does give a range for the true population proportion, so this statement is true.

05

Analyzing Statement (c)

Statement (c) discusses the nature of confidence intervals. It is true because 95% confidence means that, in repeated sampling, 95% of intervals would contain the true population proportion.

06

Analyzing Statement (d)

Statement (d) asks about changing the margin of error. To halve the margin of error, the sample size must be quadrupled. Since \( \text{margin of error} = z \sqrt{\frac{p(1-p)}{n}} \), quadrupling the sample size (\(n\)) does halve the error. Thus, this statement is true.

07

Analyzing Statement (e)

Statement (e) checks if there's evidence of a majority supporting Prop 19. Since the confidence interval (62%-78%) lies above 50%, we can conclude that a majority likely supports it. Hence, this statement is true.

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Key Concepts

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

Margin of Error

The margin of error in survey research helps us understand how much the results might differ if we were to conduct the same survey multiple times. It is a cushion around the sampled data to give an estimated range of where the real answer lies in the population. Think of it like a safety net that accounts for the tiny inconsistencies that might occur in a sample.

In the context of our example, the margin of error is 8%. This means that while 70% of our sample supports Prop 19, the true level of support in the larger population could reasonably be as low as 62% or as high as 78%.

When we say a result has a margin of error, it essentially tells us the extent to which we can expect the results to fluctuate if we take another sample under the same conditions.

Sample Proportion

The sample proportion is the percentage of people in the sample who have a particular characteristic—in our case, the number of respondents supporting Prop 19. Calculating it is fairly simple. You just need to divide the number of favorable outcomes by the total number of respondents in the sample.

In our survey, 70% of the respondents said they would vote for Prop 19. This is the sample proportion, denoted usually as \( \hat{p} \). It serves as a basis to estimate the population parameter, helping us infer what percentage of all voters might support the proposition.

The sample proportion is fundamental in calculating confidence intervals and understanding the probable reality within the whole population.

Population Proportion

The population proportion is the percentage of all individuals in the population who possess a particular attribute—in this case, support for Prop 19. Unlike the sample proportion, this is what we're ultimately trying to infer from our sample data.

The true population proportion is unknown, and that’s why statistical methods like confidence intervals exist. They help us make educated guesses about this value by using information obtained from the sample.

In our scenario, we're using the sample proportion (70%) and adding or subtracting the margin of error to estimate a range—between 62% and 78%—where the true population proportion likely falls. This practical technique allows researchers to draw meaningful conclusions from samples while considering potential variability.

Sample Size

Sample size refers to the number of respondents or observations included in a survey. Its size plays a crucial role in determining the reliability of the estimates we make about the population.

In this example, we have a sample size of 119. Larger sample sizes generally provide more accurate estimates because they tend to approximate the population more closely. This impacts the margin of error—larger samples typically result in smaller margins of error, allowing for tighter, more precise confidence intervals.

For instance, to reduce the margin of error from 8% to 4%, statement (d) explains that we would need to quadruple our sample size. This is because the margin of error is inversely proportional to the square root of the sample size. Larger samples lead to better estimations of the population proportion.