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Adult Americans (18 years or older) were asked whether they used social media (Facebook, Twitter, and so on ) regularly. The following table is based on the results of the survey. $$ \begin{array}{lccccc} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \begin{array}{l} \text { Use social } \\ \text { media } \end{array} & 117 & 89 & 83 & 49 & \mathbf{3 3 8} \\ \hline \begin{array}{l} \text { Do not use } \\ \text { social media } \end{array} & 33 & 36 & 57 & 66 & \mathbf{1 9 2} \\ \hline \text { Total } & \mathbf{1 5 0} & \mathbf{1 2 5} & \mathbf{1 4 0} & \mathbf{1 1 5} & \mathbf{5 3 0} \\ \hline \end{array} $$ (a) What is the probability that a randomly selected adult American uses social media, given the individual is \(18-34\) years of age? (b) What is the probability that a randomly selected adult American is \(18-34\) years of age, given the individual uses social media? (c) Are 18 - to 34 -year olds more likely to use social media than individuals in general? Why?

Step by step solution

01

(a) Find Probability of Social Media Use Given Age 18-34

First, identify the numbers related to adults aged 18-34 who use social media. From the table, 117 adults aged 18-34 use social media out of a total of 150 in this age group. The conditional probability formula is \[ P(\text{Use social media} \, | \, \text{Age 18-34}) = \frac{\text{Number of 18-34 year-olds who use social media}}{\text{Total number of 18-34 year-olds}} \] So, \[ P(\text{Use social media} \, | \, \text{Age 18-34}) = \frac{117}{150} = 0.78 \]

02

(b) Find Probability of Age 18-34 Given Social Media Use

Identify the numbers related to social media use and being aged 18-34. From the table, 117 adults use social media in the 18-34 age group out of a total of 338 social media users. The conditional probability formula here is \[ P(\text{Age 18-34} \, | \, \text{Use social media}) = \frac{\text{Number of 18-34 year-olds who use social media}}{\text{Total number of social media users}} \] So, \[ P(\text{Age 18-34} \, | \, \text{Use social media}) = \frac{117}{338} \approx 0.346 \]

03

(c) Compare Likelihood of Social Media Use for 18-34 Year-Olds vs. General Population

Compare the probability of social media use for the 18-34 age group to the general population. From part (a), the probability for 18-34 year-olds is 0.78. For the general population, the probability is \[ P(\text{Use social media}) = \frac{\text{Total number of social media users}}{\text{Total adult population}} \] So, \[ P(\text{Use social media}) = \frac{338}{530} \approx 0.638 \]Since 0.78 (18-34 year-olds' probability) is greater than 0.638 (general population's probability), 18-34 year-olds are more likely to use social media.

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Key Concepts

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

Probability and Conditional Probability

Probability is a fundamental idea in statistics. It measures the likelihood that an event will happen. For instance, when flipping a coin, the probability of landing on heads is 50%. When dealing with multiple conditions, conditional probability comes into play. Conditional probability helps identify the chances of an event occurring given that another event has already happened. It's calculated using the formula: \(P(A | B) = \frac{P(A \text{ and } B)}{P(B)}\) For example, in our exercise, \(P(\text{Use social media} \text{ | Age 18-34}) = \frac{117}{150} = 0.78\). This signifies that if we randomly select someone aged 18-34, there's a 78% chance they use social media.

Social Media Usage and Age Groups

Social media usage varies significantly across different age groups. In our exercise, we examined data from various age categories: 18-34, 35-44, 45-54, and 55+. The younger demographic, especially those aged 18-34, are typically more active on platforms like Facebook and Twitter. From the table, 117 out of 150 individuals in this age group use social media, making the usage probability 0.78. In contrast, among those 55+, only 49 out of 115 use social media, which is much lower. Such trends indicate how tech-savvy younger generations are compared to older ones. These insights help companies and researchers understand target audiences better.

Statistical Analysis in Surveys

Statistical analysis is crucial in interpreting survey data. This exercise involved analyzing survey information on social media use among different age groups. By breaking down and comparing probabilities for specific groups, like the 18-34 age group versus the general population, we draw meaningful conclusions. For instance, by calculating that the probability of social media usage in the general population is about 63.8% (338 out of 530), we see significant usage across all ages. Comparing this to the 78% probability for 18-34 year-olds, it's evident that younger adults are more inclined towards social media. Such analyses guide businesses, policymakers, and researchers, providing strategies and insights based on actual data.