91Ó°ÊÓ

A 2017 survey of U.S. adults found that \(74 \%\) believed that protecting the rights of those with unpopular views is a very important component of a strong democracy. Assume the sample size was 1000 . a. How many people in the sample felt this way? b. Is the sample large enough to apply the Central Limit Theorem? Explain. Assume all other conditions are met. c. Find a \(95 \%\) confidence interval for the proportion of U.S. adults who believe that protecting the rights of those with unpopular views is a very important component of a strong democracy. d. Find the width of the \(95 \%\) confidence interval. Round your answer to the nearest tenth percent. e. Now assume the sample size was 4000 and the percentage was still \(74 \%\). Find a \(95 \%\) confidence interval and report the width of the interval. f. What happened to the width of the confidence interval when the sample size was increased? Did it increase or decrease?

Step by step solution

01

Calculation of absolute numbers

Since 74% of 1000 people are 740, 740 people in the sample believe that protecting the rights of those with unpopular views is very important.

02

Application of the Central Limit Theorem

The Central Limit Theorem (CLT) generally applies when the sample size is greater than 30. Given a sample size of 1000, the CLT applies.

03

Determination of the 95% confidence interval

The standard error for a proportion is computed by \(\sqrt{(p(1-p)/n)}\), where \(p\) is the sample proportion (0.74) and \(n\) is the sample size (1000). Using the standard z-value of 1.96 for a 95% confidence level, the confidence interval can be calculated as \(p \pm z*(\sqrt{(p(1-p)/n)})\).

04

Calculation and comparison of widths of confidence intervals

The width of a confidence interval is the upper limit minus the lower limit. Compute this for the 95% confidence interval calculated in the previous step and round to the nearest tenth. Then repeat the process for a larger sample size of 4000.

05

Effect of sample size on the width of confidence intervals

Explain that as the sample size increases, the width of the confidence interval decreases, implying that the estimate becomes more precise.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values used to estimate a population parameter. In this context, it is used to estimate the proportion of U.S. adults who believe that protecting unpopular views is crucial to a strong democracy. The confidence interval gives us an idea of the uncertainty around this estimate. Usually, a confidence interval is associated with a confidence level, like 95%, which means we expect the true parameter to fall within this range 95% of the time if we repeated the survey many times.

• The formula to calculate a confidence interval for a proportion is: \[ p \pm z \times \sqrt{\frac{p(1-p)}{n}} \]
• Here, \(p\) is the sample proportion, \(z\) is the z-value corresponding to the confidence level, and \(n\) is the sample size.

Confidence intervals become narrower as the sample size increases or when we decrease the confidence level, leading to a more precise estimate.

Sample Size

Sample size is a crucial component in survey sampling and statistical analysis. It determines the accuracy and reliability of the results. In the exercise, we see two different scenarios with sample sizes of 1000 and 4000.

• A larger sample size, like 4000, reduces the standard error, which narrows the confidence interval width. This means the estimate of the proportion is more precise.
• For applying the Central Limit Theorem (CLT), a sample size greater than 30 is often sufficient, which is easily met in this exercise.

Thus, increasing the sample size enhances the survey's reliability while lowering the margin of error in the confidence interval.

Statistical Proportion

Statistical proportion refers to the part of the total that exhibits a particular attribute, represented as a fraction or percentage. In this problem, it pertains to the 74% of surveyed U.S. adults.

• The sample proportion \( p \) is computed by dividing the number of people expressing a particular view (740 in this exercise) by the total sample size (1000).
• Proportions lie between 0 and 1 or 0% to 100%, indicating the proportion of the sample that meets the specific criteria.

Calculating the sample proportion can help estimate the population proportion when combined with methods like confidence intervals and tests involving the z-value.

Z-value

The z-value, in statistics, is a standard score that represents the number of standard deviations a data point is from the mean. In forming a confidence interval, it helps decide how far away the interval extends from the sample proportion.

• A common z-value used for a 95% confidence interval is 1.96, which assumes a normal distribution.
• The z-value corresponds to the level of confidence chosen, reflecting the probability that the true population parameter lies within the confidence interval.
• A higher z-value indicates greater confidence, but results in a wider confidence interval.

This balancing act of z-value and interval width illustrates the trade-off between confidence and precision in statistical analysis.