Question

Information from a poll of registered voters in Cedar Rapids, Iowa, to assess voter support for a new school tax was the basis for the following statements (Cedar Rapids Gazette, August 28,1999 ): The poll showed 51 percent of the respondents in the Cedar Rapids school district are in favor of the tax. The approval rating rises to 56 percent for those with children in public schools. It falls to 45 percent for those with no children in public schools. The older the respondent, the less favorable the view of the proposed tax: 36 percent of those over age 56 said they would vote for the tax compared with 72 percent of 18- to 25 -year-olds. Suppose that a registered voter from Cedar Rapids is selected at random, and define the following events: \(F=\) event that the selected individual favors the school \(\operatorname{tax}, C=\) event that the selected individual has children in the public schools, \(O=\) event that the selected individual is over 56 years old, and \(Y=\) event that the selected individual is \(18-25\) years old. a. Use the given information to estimate the values of the following probabilities: i. \(P(F)\) ii. \(P(F \mid C)\) iii. \(P\left(F \mid C^{C}\right)\) iv. \(P(F \mid O)\) v. \(P(F \mid Y)\) b. Are \(F\) and \(C\) independent? Justify your answer. c. Are \(F\) and \(O\) independent? Justify your answer.

Short answer

i. \(P(F)\) = 0.51 \nii. \(P(F|C)\) = 0.56 \niii. \(P(F|C^C)\) = 0.45 \niv. \(P(F|O)\) = 0.36 \nv. \(P(F|Y)\) = 0.72 \nb. F and C are not independent. \nc. F and O are not independent.

Step-by-step solution

Identify the given information

The following information is provided: \n\n1. 51% of respondents favor the tax. \n2. The approval rating rises to 56% for those with children in public schools. \n3. The approval rating falls to 45% for those with no children in public schools. \n4. 36% of those over age 56 said they would vote for the tax. \n5. 72% of 18 - to 25 -year-olds said they would vote for the tax.

Calculate the individual probabilities

From the given information, we can directly determine: \n\ni. \(P(F)\) = 0.51 (probability that a randomly chosen individual supports the tax) \nii. \(P(F|C)\) = 0.56 (probability that a randomly chosen individual supports the tax given that they have children in public schools) \niii. \(P(F|C^C)\) = 0.45 (probability that an individual supports the tax given that they don't have children in public schools) \niv. \(P(F|O)\) = 0.36 (probability that a randomly chosen individual supports the tax given they are over the age of 56) \nv. \(P(F|Y)\) = 0.72 (probability that an individual supports the tax given they are between 18 and 25 years old)

Determine independence of F and C

Two events are independent if the occurrence of one does not change the probability of the other. Mathematically, two events A and B are independent if \(P(A|B) = P(A)\). So to determine if F and C are independent, we can compare \(P(F|C)\) to \(P(F)\). So if 0.56 (which is \(P(F|C)\)) equals 0.51 (which is \(P(F)\)), the two events are independent. Since these values are not equal, F and C are not independent.

Determine independence of F and O

Following the same reasoning as in the previous step, we compare \(P(F|O)\) to \(P(F)\). So if 0.36 (which is \(P(F|O)\)) equals 0.51 (which is \(P(F)\)), the two events are independent. Since these values are not equal, F and O are not independent.