7.
In a certain survey, 30% of those questioned are given an incentive to complete the survey. Of those who receive incentives, 80% completed the survey. Of those who do not receive incentives, 25% completed the survey.
a. Find the probability of getting incentives for those who completed the survey. Hint: Use Bayes’ Theorem
b. If one questioned subject is selected at random, find the probability that is completed. Hint: The answer is in part a.
I found the part A answer to be 0.578 by using bayes.
(0.8*0.3)/(0.8*0.3 + 0.25*0.7)
but i have no idea on how to find b. I thought it woudl be .8*.25 because .8 completed the survey with incentives and .25 completed survey so the proabiblity would be 1/(.8*.25).
a. agree
b The first term in the denominator is those who got the incentive and completed the survey.
The second term in the denominator is those who completed the survey without the incentive.
The sum is all those who completed the survey, which is what you are looking for.
To find part b, the probability that a questioned subject completed the survey, we need to consider both those who received incentives and those who did not.
Let's break down the question:
1. 30% of those questioned are given an incentive to complete the survey.
2. Among those who received incentives, 80% completed the survey.
3. Among those who did not receive incentives, 25% completed the survey.
Since we know that those who received incentives represent 30% of the total, and among them, 80% completed the survey, we can calculate the probability of a questioned subject completing the survey given that they received incentives as follows:
Probability of completing the survey with incentives = 0.80 * 0.30 = 0.24 (24%)
Similarly, for those who did not receive incentives, since they represent 70% of the total and only 25% of them completed the survey, we can calculate the probability of a questioned subject completing the survey without incentives:
Probability of completing the survey without incentives = 0.25 * 0.70 = 0.175 (17.5%)
Now, to calculate the overall probability of a questioned subject completing the survey, we need to take into account the probability of receiving incentives and the probability of completing the survey given incentives, as well as the probability of not receiving incentives and the probability of completing the survey without incentives.
Probability of completing the survey =
(Probability of completing the survey with incentives * Probability of receiving incentives) +
(Probability of completing the survey without incentives * Probability of not receiving incentives) =
(0.24 * 0.30) + (0.175 * 0.70) =
0.072 + 0.1225 =
0.1945 (19.45%)
Therefore, the probability that a questioned subject completed the survey is 0.1945 or 19.45%.
The formula you mentioned, 1/(0.8 * 0.25), does not appropriately calculate the probability in this case because it assumes that the completion rate is directly related to receiving incentives, disregarding the different completion rates for those who received incentives and those who did not.