Question 1192535: Higher education at two-year and four-year institutions. The following table gives the counts of U.S. institutions of higher education classified as public or private and as two-year or four-year:
Public Private
Two-year 1000 721
Four-year 2774 672
Convert the counts to probabilities and summarize the relationship between these two variables using conditional probabilities.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Calculate the Total Number of Institutions**
* Total institutions = 1000 (Public, Two-year) + 721 (Private, Two-year) + 2774 (Public, Four-year) + 672 (Private, Four-year)
* Total institutions = 5167
**2. Calculate Probabilities**
* **P(Public)** = (Public Two-year + Public Four-year) / Total = (1000 + 2774) / 5167 = 0.721
* **P(Private)** = (Private Two-year + Private Four-year) / Total = (721 + 672) / 5167 = 0.279
* **P(Two-year)** = (Public Two-year + Private Two-year) / Total = (1000 + 721) / 5167 = 0.331
* **P(Four-year)** = (Public Four-year + Private Four-year) / Total = (2774 + 672) / 5167 = 0.669
* **P(Public and Two-year)** = 1000 / 5167 = 0.193
* **P(Public and Four-year)** = 2774 / 5167 = 0.538
* **P(Private and Two-year)** = 721 / 5167 = 0.139
* **P(Private and Four-year)** = 672 / 5167 = 0.130
**3. Calculate Conditional Probabilities**
* **P(Two-year | Public)** = P(Public and Two-year) / P(Public) = 0.193 / 0.721 = 0.268
* **P(Four-year | Public)** = P(Public and Four-year) / P(Public) = 0.538 / 0.721 = 0.747
* **P(Two-year | Private)** = P(Private and Two-year) / P(Private) = 0.139 / 0.279 = 0.498
* **P(Four-year | Private)** = P(Private and Four-year) / P(Private) = 0.130 / 0.279 = 0.466
**Summary of the Relationship**
* **Public institutions are more likely to be four-year institutions** (74.7%) than two-year institutions (26.8%).
* **Private institutions are more likely to be two-year institutions** (49.8%) than four-year institutions (46.6%).
This analysis suggests a relationship between the type of institution (public vs. private) and the level of education (two-year vs. four-year).
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