In politics, marketing, etc. we often want to estimate a percentage or proportion $p$. One calculation in statistical polling is the margin of error - the largest (reasonble) error that the poll could have. For example, a poll result of $72 \%$ with a margin of error of $4 \%$ indicates that $p$ is most likely to be between $68 \%$ and $76 \%(72 \%$ minus $4 \%$ to $72 \%$ plus $4 \%)$. In a (made-up) poll, the proportion of people who like dark chocolate more than milk chocolate was $41 \%$ with a margin of error of $1.9 \%$. Describe the conclusion about $p$ using an absolute value inequality. The answer field below uses the symbolic entry option in Mobius. That lets you type in a vertical bar | to represent absolute values. Also, when you type in $<$ and then $=$, the symbolic entry option will automatically convert that to $\leq$. In the same way, if you type in $>$ and then $=$, the symbolic entry option will automatically convert that to $\geq$. Be sure to use decimal numbers in your answer (such as using 0.40 for $40 \%$ ).
Solution
Step 1 ：The question is asking for an absolute value inequality that describes the conclusion about the proportion $p$ of people who like dark chocolate more than milk chocolate. The proportion was found to be $41 \%$ with a margin of error of $1.9 \%$.
Step 2 ：This means that the true proportion $p$ is most likely to be between $41 \% - 1.9 \%$ and $41 \% + 1.9 \%$. In decimal form, this is between $0.41 - 0.019$ and $0.41 + 0.019$.
Step 3 ：We can express this as an absolute value inequality in the form $|p - a| \leq b$, where $a$ is the observed proportion and $b$ is the margin of error.
Step 4 ：In this case, $a = 0.41$ and $b = 0.019$. So the inequality is $|p - 0.41| \leq 0.019$.
Step 5 ：This inequality says that the difference between the true proportion $p$ and the observed proportion $0.41$ is at most $0.019$, which is the margin of error.
Step 6 ：Final Answer: \(\boxed{|p - 0.41| \leq 0.019}\)
