Question

An economist wants to study the impact of free financial counseling on saving behavior. The economist has a small random sample of people who have been “rolling over” their credit card debt for six months (i.e. they have not been paying the balance on their credit card, and therefore interest charges have been accumulating). All of these people receive financial counseling. The economist will observe whether or not each of them pays off their credit card debt within six months. In historical data, approximately 0.8 of people in this situation paid off their credit card debt within six months even without financial counseling. Therefore, the economist wants to test the null hypothesis that with financial counseling, the probability p that a person who rolls over their credit card debt for six months will pay it off within six months is less than or equal to 0.8. The alternative hypothesis is that the probability is greater than 0.8. Use 5 decimals in your calculations even though you are asked for 4 decimals in your answers. Suppose the sample size is 15. The economist adopts the rejection region: “Reject Ho if all 15 people in the sample pay off their credit card debt within six months”, i.e. “Reject Ho if X > cv, where cv = 14". Calculate the probability that Ho will be rejected if p = 0.8. Four decimals

Answer 1

Null hypothesis, H₀: p $\leq$ 0.8 Alternative hypothesis, H₁: p>0.8.

Prob 1: P(X>14|p=0.8)=0.0352 R code: round(dbinom(15,15,0.8),4)

Option: Probability of Type I error.

Prob 2: P(X>14|p=0.75)=0.0134 R code: round(dbinom(15,15,0.75),4)

Option: Probability of Type I error.

Prob 3: P(X>14|p=0.85)=0.0874 R code: round(dbinom(15,15,0.85),4)

Option: Power of the test.

Prob 4: P(X>14|p=0.90)=0.2059 R code: round(dbinom(15,15,0.90),4)

Option: Power of the test.

Prob 5: As p increases the P(X>14|p) increases

Option: Positive slope (upward-sloping from left to right)