91Ó°ÊÓ

Baseball fans In a national poll taken in February 2008 Gallup asked 1006 adults whether they were baseball fans; \(43 \%\) said they were. Two months previously, in December \(2007,40 \%\) of a similar-size sample had reported being baseball fans. a) Find the margin of error for the 2008 poll if we want \(90 \%\) confidence in our estimate of the percent of national adults who are baseball fans. b) Explain what that margin of error means. c) If we wanted to be \(99 \%\) confident, would the margin of error be larger or smaller? Explain. d) Find that margin of error. e) In general, if all other aspects of the situation remain the same, will smaller margins of error produce greater or less confidence in the interval?

Step by step solution

01

Understanding the Margin of Error Formula

The margin of error (MOE) for a proportion in a poll can be calculated using the formula: \[ MOE = z \times \sqrt{\frac{p(1-p)}{n}} \] where \( z \) is the critical value from the z-table corresponding to the desired confidence level, \( p \) is the estimated proportion, and \( n \) is the sample size.

02

Calculate Margin of Error for 90% Confidence

For 90% confidence, the z-value is approximately 1.645. Using \( p = 0.43 \) and \( n = 1006 \), the margin of error is calculated as follows: \[ MOE = 1.645 \times \sqrt{\frac{0.43(1-0.43)}{1006}} \approx 0.025 \] Thus, the margin of error for this poll is approximately 2.5%.

03

Interpret the Margin of Error

The margin of error of 2.5% means that we can be 90% confident that the true proportion of adults who are baseball fans in the 2008 poll is within 2.5 percentage points of the reported 43%.

04

Compare Margin of Errors for Different Confidence Levels

For a higher confidence level, such as 99%, the critical value \( z \) will be larger, approximately 2.576. Therefore, the margin of error will increase when the confidence percentage increases.

05

Calculate Margin of Error for 99% Confidence

Using the same formula as before but with a z-value of 2.576, the margin of error is: \[ MOE = 2.576 \times \sqrt{\frac{0.43(1-0.43)}{1006}} \approx 0.039 \] Hence, the margin of error for 99% confidence is approximately 3.9%.

06

Analyze Margin of Error versus Confidence Relationship

In general, a smaller margin of error corresponds to less confidence in the interval because a tighter interval requires more surety about where the true population proportion lies. Conversely, increasing confidence typically results in a larger margin of error to ensure the population parameter is captured by the interval.

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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 provides a range of values that likely contain the true value of the parameter being measured, in this case, the proportion of baseball fans among adults. When we say we are 90% confident, it means that if we were to take many samples and create confidence intervals for each, approximately 90% of those intervals would contain the true population proportion.
Confidence intervals are useful because they give us a sense of the uncertainty surrounding our estimate.

• A wider confidence interval means more uncertainty, whereas a narrower one means less uncertainty.
• To compute a confidence interval, you need the estimated proportion, the critical value (z-score), and the sample size.

Understanding this concept is fundamental when analyzing survey data, like in the national poll discussed, to make informed decisions based on the data.

Proportion Calculation

Proportion calculation is an essential step in estimating how many members of a population have a certain characteristic, here, being baseball fans. In the exercise, the poll result showed that 43% of respondents identified as fans.
The formula for calculating the margin of error ( (MOE)) uses this proportion value.
The calculated proportion is crucial for designing the confidence interval:

• The proportion of 0.43 (or 43%) serves as the center of our confidence interval.
• It remains fixed when calculating intervals across different confidence levels, while the critical value can change.
• Other variables like sample size also influence how accurately we can trust our proportion estimate.

Knowing how to calculate and utilize proportions helps in understanding broader concepts like the margin of error and ensures that the data accurately reflects reality.

Critical Value

The critical value in statistics is a factor used to compute the margin of error that reflects how confident we are in our estimates. It is derived from the standard normal distribution and is determined by the chosen confidence level for the interval.
In the exercise, different z-values are given for distinct confidence levels:

• For a 90% confidence level, the critical value is approximately 1.645.
• For a 99% confidence level, the critical value increases to about 2.576.

This means, as confidence increases, the critical value used in the margin of error formula also increases.
Understanding critical values is important as they adjust the width of the confidence interval. Greater critical values expand the confidence region, thus increasing certainty that the interval will contain the true population parameter. However, this also increases the margin of error.

Poll Analysis

Poll analysis involves interpreting the data collected from surveys to make inferences about a larger population. Here, the national poll assesses how many adults identify as baseball fans.
Effective poll analysis requires accurately calculating confidence intervals and margin of error, as seen in this exercise:

• Poll results are subject to sampling variability, hence the need for margin of error to account for potential discrepancies.
• By analyzing the poll through mathematical tools, we can infer that between about 40.5% and 45.5% of adults could be baseball fans with 90% confidence.
• Adjusting confidence levels for greater certainty affects how poll results are interpreted, requiring deeper analysis.

Therefore, good poll analysis is not just about collecting data, but also about understanding and properly interpreting it using statistical methods to ensure accurate outcomes and informed conclusions.