Question

What Proportion of Adults and Teens Text Message? A study of \(n=2252\) adults age 18 or older found that \(72 \%\) of the cell phone users send and receive text messages. \({ }^{15}\) A study of \(n=800\) teens age 12 to 17 found that \(87 \%\) of the teen cell phone users send and receive text messages. What is the best estimate for the difference in the proportion of cell phone users who use text messages, between adults (defined as 18 and over) and teens? Give notation (as a difference with a minus sign) for the quantity we are trying to estimate, notation for the quantity that gives the best estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Short answer

The estimated difference in the proportion of adults and teens who text is -0.15 (-indicating a higher proportion of teens). The target quantity is denoted as \( D = P_{a} - P_{t} \) and the best estimate for this quantity is denoted as \( d = p_{a} - p_{t} \), with \( p_{a} = 0.72 \) and \( p_{t} = 0.87 \) being the observed sample proportions of adults and teens respectively.

Step-by-step solution

Identify the Variables

We have two variables to consider: the proportion of adults and the proportion of teens who send and receive text messages. Let's denote the population proportion of adults who text as \( P_{a} \) and the population proportion of teens who text as \( P_{t} \). Our goal is to estimat1e the difference \( D = P_{a} - P_{t} \). The given data provides us with estimates for these proportions, known as sample proportions: for adults, \( p_{a} = 0.72 \), and for teens, \( p_{t} = 0.87 \).

Calculate the Difference in Proportions

To estimate the difference between the two proportions, we simply subtract the sample proportion of the adults from the sample proportion of the teens. This results in the sample estimate of the difference \( d = p_{a} - p_{t} = 0.72 - 0.87 = -0.15 \). This negative value indicates that the proportion of teens who text is higher than the proportion of adults who text.

Explain the Notations and the Parameters

Let's now clarify the notations: \( P_{a} \) and \( P_{t} \) are the population proportions of adults and teens (respectively) who use text messages -- these are parameters we'd like to know, but don't have enough information to find exactly. \( p_{a} = 0.72 \) and \( p_{t} = 0.87 \) are the sample proportions of adults and teens who text -- these are estimates based on the sample data. \( D = P_{a} - P_{t} \) is the difference in population proportions, and \( d = -0.15 \) is the observed difference in sample proportions -- this is our best estimate for D in this situation.