91Ó°ÊÓ

In a study of the scientific research on soft drinks, juices, and milk, 50 studies were fully sponsored by the food industry, and 30 studies were conducted with no corporate ties. Of those that were fully sponsored by the food industry, \(14 \%\) of the participants found the products unfavorable, \(23 \%\) were neutral, and \(63 \%\) 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 participant selected at random found the product favorable, what is the probability that he or she belongs to a group that participated in a corporate-sponsored study?

Step by step solution

01

Decode the given information

We have 50 studies sponsored by the food industry and 30 studies without corporate ties. We need to find the total number of participants for both groups. We are given the percentage of people in each group that had certain opinions, so we can calculate the number of participants with those opinions for both groups.

02

Calculate the number of favorable opinions in both groups

We need to find the total number of favorable opinions in both groups: Sponsored group: Number of favorable opinions: \(50\times 0.63 = 31.5 \approx 32\) (rounding to the nearest whole number) Non-sponsored group: Number of favorable opinions: \(30\times 0.47 = 14.1 \approx 14\) (rounding to the nearest whole number)

03

Calculate the probability of a participant finding the product favorable

We are now ready to calculate the probability of a participant finding the product favorable (part a): Total number of participants: \(50 + 30 = 80\) Total number of favorable opinions: \(32 + 14 = 46\) Probability that a participant found the products favorable: \( \frac{46}{80} = 0.575 \) a. The probability that a participant selected at random found the products favorable is \(57.5\%\).

04

Calculating the probabilities of belonging to each group

For part b, we will use Bayes' theorem. First, we need to calculate the probability of a participant belonging to the sponsored or the non-sponsored group: \(P(Sponsored) = \frac{50}{80} = 0.625\) \(P(Non-Sponsored) = \frac{30}{80} = 0.375\)

05

Calculating the probability of a participant finding the product favorable given they belong to the sponsored group

We also need the probability of finding the product favorable given they belong to the sponsored group: \(P(Favorable|Sponsored) = \frac{32}{50} = 0.64\)

06

Use Bayes' theorem to find the probability of being in a corporate-sponsored study given a favorable opinion

We will use Bayes' theorem to calculate the probability that a participant who found the product favorable belongs to a corporate-sponsored study group: \(P(Sponsored|Favorable) = \frac{P(Favorable|Sponsored) \times P(Sponsored)}{P(Favorable)}\) \(P(Sponsored|Favorable) = \frac{0.64 \times 0.625}{0.575} \approx 0.696\) b. If a participant selected at random found the product favorable, the probability that he or she belongs to a group that participated in a corporate-sponsored study is approximately \(69.6\%\).

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!