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In June 2008 , SBRI Public Affairs conducted a telephone poll of 1004 adult Americans aged 18 and older. One of the questions asked was, "In the past year, was there ever a time when you ...?" Respondents could choose more than one of the answers mentioned. Of the respondents, \(64 \%\) said "cut back on vacations or entertainment because of their cost," \(37 \%\) said "failed to pay a bill on time," and \(25 \%\) said "have not gone to a doctor because of the cost." (Source: http://www.srbi.com/AmericansConcernEconomic.html.) Using these results, find a \(95 \%\) confidence interval for the corresponding population percentage for each answer. Write a one-page report to present these results to a group of college students who have not taken statistics. Your report should answer questions such as: (1) What is a confidence interval? (2) Why is a range of values more informative than a single percentage? (3) What does \(95 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?

Step by step solution

01

Understand Confidence Interval

A confidence interval in statistics gives an estimated range of values that is likely to contain an unknown population parameter. It adds a margin of error around the point estimate to give a range within which the population parameter will fall.

02

Value of a Range Vs Single Percentage

A range of values is more informative than a single percentage as it represents variability and uncertainty. It showcases the margin of error, the minimum and maximum values which give a broader picture than a single point estimate.

03

Define 95% Confidence

A 95% confidence interval means, if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value(µ).

04

Calculating Confidence Interval

The 95% confidence interval for a sample population can be calculated using the formula \(CI= \hat{p} ± Z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). Here, \(\hat{p}\) represents the sample proportion and n is the sample size. Use this formula to calculate the confidence interval for each of the answers given in the poll.

05

Making Assumptions

Assumptions made when constructing a confidence interval include: the data was collected randomly, the sample distribution approximates a normal distribution, and observations are independent of each other.

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Key Concepts

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

Sample Proportion

The sample proportion is a key concept in statistics, especially when dealing with survey data. It represents the portion or percentage of the sample that exhibits a particular characteristic. For instance, if 64% of poll respondents said they cut back on vacations due to cost, the sample proportion for this behavior is 0.64.

The sample proportion helps us estimate the population proportion, which is what we aim to understand through the poll. Although the sample is a subset, it gives a glimpse into the habits or opinions of the entire population.

Understanding the sample proportion is vital because it serves as a foundation for calculating the confidence interval. It blends collected data with statistical theory to provide insights into broader patterns.

• Represents the proportion in a sample exhibiting a characteristic
• Helps estimate the population proportion
• Is a foundation for the confidence interval calculation

Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It appears as a plus-and-minus in a confidence interval and represents the maximum expected difference between the true population parameter and a sample estimate.

For example, if we say that 64% of the population cut back on vacations with a margin of error of 3%, the actual population percentage could realistically be between 61% and 67%.

Why is this important? It indicates the expected range of fluctuation in our estimates, revealing the uncertainty inherent in using a sample to draw conclusions about a larger population.

• Shows possible error in survey results
• Indicates the confidence interval’s range around the sample estimate
• Highlights uncertainty in estimating population characteristics

Population Parameter

A population parameter is a value that represents a characteristic of an entire population, and it is the true proportion we want to estimate using our sample data. In the context of the poll, these parameters might include the true percentage of all adult Americans who cut back on vacations or entertainment.

Unlike sample statistics, population parameters are fixed yet unknown, and they provide benchmarks against which we measure our sample estimates. By understanding these parameters, statisticians can make educated guesses about the behavior or traits of the entire group under study.

In our case, confidence intervals are used to estimate these parameters, giving a range of plausible values rather than a single point.

• Represents a full characteristic of the population
• Estimated from the sample to offer insights into the population
• Used with confidence intervals for broader understanding

95% Confidence Level

A 95% confidence level means that if we were to take 100 different samples and calculate a confidence interval for each one, about 95 of those intervals would contain the true population parameter. It is a statistical standard that signifies the degree of certainty we have in our estimation.

This concept can be a bit challenging, but think of it as a way to express reliability. The higher the confidence level, the more assurance we have. However, it often comes with a wider margin of error.

Why use a 95% confidence level? It represents a good balance between certainty and practicality, commonly accepted in social science and related fields.

• Indicates reliability of confidence interval
• 95 out of 100 samples should capture the actual parameter
• Widely used for practical and reliable results