Reading Books A HuffPost/YouGov poll conducted September...

Reading Books A HuffPost/YouGov poll conducted September $27-28,2013$, found that Americans read an average of 13.6 books per year. Suppose HuffPost/YouGov is 99\% confident that the result from this poll is off by fewer than 1.8 books from the actual average $x .$ Express this situation as an inequality involving absolute value, and solve the inequality for $x$ to determine the interval in which the actual average is likely to fall. [Note: In statistics, this interval is called a $99 \%$ confidence interval.]

Key Concepts

Absolute Value Inequality

An absolute value inequality is used to represent the notion of distance between two quantities on the number line. For instance, expressing that the difference between an observed statistic and the population parameter is less than a certain amount, like 1.8 in this case, is written as |observed value ? parameter| < margin of error. This formulation captures the idea that the true value is within a specific range around the observed statistic.

Solving Absolute Value Inequalities

To solve an absolute value inequality such as |x ? a| < b, one converts it into the compound inequality a ? b < x < a + b. This method allows for finding the interval in which the variable x lies, and it is a fundamental technique when determining a range of plausible values for a parameter based on a sample statistic.

Confidence Interval

A confidence interval in statistics is a range of values, derived from sample data, that is believed to contain the true population parameter with a specified level of confidence (such as 99%). It provides not only an estimate of the parameter but also an indication of the reliability of the estimate by incorporating the margin of error.

Margin of Error

The margin of error represents the maximum expected difference between the sample statistic and the true population parameter, based on the confidence level selected. It quantifies the uncertainty inherent in the sampling process and is used to determine the width of the confidence interval.

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