SOLUTION: Scientific research on popular beverages consisted of 60 studies that were fully sponsored by the food industry, and 40 studies that were conducted with no corporate ties. Of those

Question 1168489: Scientific research on popular beverages consisted of 60 studies that were fully sponsored by the food industry, and 40 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 15 % of the participants found the products unfavorable, 23 % were neutral, and 62 % found the products favorable. Of those that had no industry funding, 38 % found the products unfavorable, 15 % were neutral, and 47 % found the products favorable.
a)What is the probability that a participant selected at random found the products favorable?
b)If a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry?
c)If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's break down this problem step by step.
**1. Organize the Data**
* **Industry Sponsored (IS):** 60 studies
* Unfavorable (U_IS): 15% of 60 = 0.15 * 60 = 9 studies
* Neutral (N_IS): 23% of 60 = 0.23 * 60 = 13.8 studies (round to 14)
* Favorable (F_IS): 62% of 60 = 0.62 * 60 = 37.2 studies (round to 37)
* **No Industry Funding (NI):** 40 studies
* Unfavorable (U_NI): 38% of 40 = 0.38 * 40 = 15.2 studies (round to 15)
* Neutral (N_NI): 15% of 40 = 0.15 * 40 = 6 studies
* Favorable (F_NI): 47% of 40 = 0.47 * 40 = 18.8 studies (round to 19)
* **Total Studies:** 60 + 40 = 100 studies
* **Total Favorable:** 37 + 19 = 56 studies
* **Total Unfavorable:** 9 + 15 = 24 studies
* **Total Neutral:** 14 + 6 = 20 studies
**a) What is the probability that a participant selected at random found the products favorable?**
* P(Favorable) = (Number of Favorable Studies) / (Total Studies)
* P(Favorable) = 56 / 100 = 0.56
**b) If a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry?**
* We need to find P(IS | Favorable), which is the probability of the study being industry-sponsored given that the participant found the product favorable.
* Using Bayes' Theorem:
* P(IS | Favorable) = [P(Favorable | IS) * P(IS)] / P(Favorable)
* P(Favorable | IS) = 37 / 60 ≈ 0.6167
* P(IS) = 60 / 100 = 0.6
* P(Favorable) = 0.56
* P(IS | Favorable) = (0.6167 * 0.6) / 0.56
* P(IS | Favorable) = 0.37 / 0.56
* P(IS | Favorable) ≈ 0.6607
**c) If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?**
* We need to find P(NI | Unfavorable), which is the probability of the study having no industry funding given that the participant found the product unfavorable.
* Using Bayes' Theorem:
* P(NI | Unfavorable) = [P(Unfavorable | NI) * P(NI)] / P(Unfavorable)
* P(Unfavorable | NI) = 15 / 40 = 0.375
* P(NI) = 40 / 100 = 0.4
* P(Unfavorable) = 24 / 100 = 0.24
* P(NI | Unfavorable) = (0.375 * 0.4) / 0.24
* P(NI | Unfavorable) = 0.15 / 0.24
* P(NI | Unfavorable) = 0.625
**Answers:**
a) 0.56
b) Approximately 0.6607
c) 0.625

RELATED QUESTIONS
Scientific research on popular beverages consisted of 60 studies that were fully... (answered by Boreal)

Scientific research on popular beverages consisted of 60 studies that were fully... (answered by ikleyn)

The refreshment stand at the little league field sells beverages for 1.25 each and... (answered by helpwithhomework,gonzo)

Jake's take-home pay for his first week of fruit picking was $800. His expenses for... (answered by JBarnum)

suppose a city council consisted of 7 men and 5 women. if a lobbyist for the chemical... (answered by lynnlo)

Research in the gaming industry showed that 10% of all slot machines in the United States (answered by stanbon)

b. Two research workers classified some people in income groups on the basis of sampling... (answered by stanbon)

Food Safety. The article cited research by the Food Marketing Institute, which indicates... (answered by ewatrrr)

The number of beverages introduced in the market during the two consecutive years... (answered by josmiceli)