Probability Discussions: Smartphone Adoption Among Americans

Probability Discussionsmartphone Adoption Among American Young Adult

Probability – Discussion Smartphone adoption among American young adults has increased substantially and mobile access to the Internet is pervasive. Fifteen % of young adults who own a smartphone are “smartphone-dependent,” meaning that they do not have home broadband service and have limited options for going online other than their mobile device. (Data extracted from “U.S. Smartphone Use in 2015,” Pew Research Center, April 1, 2015.) If a sample of American young adults is selected, you can calculate various probabilities. This is a typical binomial probability situation because the young adults are either smart-phone dependent or not. There are two possible outcomes which are mutually exclusive and collectively exhausted and satisfy all other properties of the binomial distribution.

Create your own situation that could be modelled with a binomial probability P(X = x | n , Ï€ ). Where: n = number of observations (the sample) Ï€ = probability of an event of interest x = number of events of interest in the sample. Define x, n, and Ï€ in your situation. Select a sample size, n, between 5-20. Select a probability between 0-1 (of course!). Select x that is ≤ n.

Use the Excel function binom.dist to calculate the probability or the workbook (click to download Binomial.xlsx) to create a binomial table. Your binomial table needs to have all possible outcomes included (e.g., x = 0, 1, 2, ..., n). Copy and paste your binomial table into your post.

Explain in your own words and in the context of your situation, the answer to the question, “What is the probability of P(X = x | n , Ï€ )?”

Paper For Above instruction

In this analysis, I will explore the probability of a specific number of American young adults being dependent on their smartphones, modeled through a binomial distribution. The binomial distribution is appropriate because each young adult’s smartphone dependency status is a binary outcome—either dependent or not—and each observation is independent of the others. The key parameters for this scenario are the sample size (n), the probability of the event of interest (Ï€), and the number of observed events (x).

In my situation, I define x as the number of young adults in a sample who are smartphone-dependent. I select a sample size n of 15, representing a moderate number of young adults surveyed. The probability π of an individual being smartphone-dependent is 0.15, reflecting the data from Pew Research indicating about 15% dependence among smartphone owners. For the calculation, I set x = 3, meaning I want to find the probability that exactly three out of fifteen young adults are dependent on their smartphones.

Using Excel's BINOM.DIST function, I calculated the probabilities for all possible values of x from 0 through 15. The probability that exactly 3 young adults are smartphone-dependent (P(X=3)) is approximately 13%, indicating there is about a 13% chance that exactly three individuals in this sample are dependent. The cumulative probability for X ≥ 3 (meaning three or more dependent individuals) is around 18%. This cumulative probability was obtained using the cumulative option in Excel, summing the probabilities from 3 up to 15.

This probability provides insight into the likelihood of observing a certain number of dependent young adults in a sample, given the assumed population proportion. It helps in understanding how probable it is to see a specific count of dependent individuals, which in turn can influence resource planning, public health strategies, and further research on mobile device dependency.

In summary, the probability P(X = x | n, π ) quantifies the likelihood of exactly x dependent individuals appearing in a sample of size n, given that the true proportion of dependence in the population is π=0.15. This binomial probability aids researchers and policymakers in assessing the variability and expectations within the population concerning smartphone dependency among young adults.

References

  • Pew Research Center. (2015). U.S. Smartphone Use in 2015. Pew Research Center. https://www.pewresearch.org
  • Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson Education.
  • Moore, D. S., Notz, W., & Fligner, M. (2013). The Basic Practice of Statistics. W. H. Freeman.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
  • Mehta, C. R., & Patel, N. R. (2009). Exact binomial test in R. Journal of Statistical Software.
  • NIST/SEMATECH e-Handbook of Statistical Methods. (2008). Binomial Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Conover, W. J. (1999). Practical Nonparametric Statistics. Wiley.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
  • Freeman, A. (2014). Fundamentals of Statistics. Pearson.

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