91Ó°ÊÓ

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.

Step by step solution

01

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, \( p_{t} = 0.87 \).

02

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.

03

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Difference of Proportions

When we talk about the difference of proportions, we're looking at how two groups compare with each other in terms of a particular characteristic. In the case of our exercise, the characteristic is the proportion of people who use text messages, and the groups are adults and teens.

• The goal is to find out how much more or less one group texts compared to the other.
• Mathematically, this is expressed as the difference between the proportions: \( D = P_{a} - P_{t} \), where \( D \) stands for the difference, \( P_{a} \) is the population proportion for adults, and \( P_{t} \) is for teens.

In our exercise, we calculated the difference using sample data, resulting in \( d = p_{a} - p_{t} = -0.15 \). This tells us that a higher proportion of teens than adults use text messaging.

Population Proportion

Now, the term population proportion might sound a bit technical, but it's important for understanding large groups. A population proportion, like \( P_{a} \) for adults or \( P_{t} \) for teens, represents the fraction or percentage of an entire group that exhibits a particular trait—in this case, the use of text messaging.

• The population proportion is a "true" value, meaning it's what we would find if we could ask every single person in the group.
• However, in practice, we don't have access to the entire population; we rely on samples to estimate these proportions.

The population proportions are what researchers ultimately want to know, but since they can only guess based on a sample, they use sample proportions.

Sample Proportion

When it's almost impossible to get answers from an entire population, researchers use a sample to estimate proportions. This is where the concept of sample proportion comes in.

• Sample proportions are denoted as \( p_{a} \) and \( p_{t} \) for adults and teens, respectively.
• These are calculated based on the sample sizes—here, 72% of 2252 adults and 87% of 800 teens.
• They serve as the best guess or estimate for the population proportions we talked about earlier.

While sample proportions are estimates, they provide valuable insights. In our exercise, they revealed that 87% of teens use text messaging compared to 72% of adults. This helps us understand the overall trend without having to survey every single person in these groups.