Think about the following game: A fair coin is tossed 10 times. Each time the to

Question

Think about the following game: A fair coin is tossed 10 times. Each time the toss results in heads, you receive $10: for tails, you get nothing. What is the maximum amount you would pay to play the game? As a statistics student, you are aware that the game is a 10-trial binomial experiment. A toss that lands on heads is defined as a success, and because the probability of a success is 0.5, the expected number of successes is 10(0.5) = 5. Since each success pays $10, the expected value of the game is 5($10) = $50. Imagine each person in a random sample of 1, 536 adults between the ages of 22 and 55 is invited to play this game. Each person is asked the maximum amount they are willing to pay to play. (Data source: These data were adapted from Ben Mansour, Selima, Jouini, Elyes, Marin, Jean-Michel, Napp, Clotilde, & Robert, Christian. (2008). Are risk-averse agents more optimistic? A Bayesian estimation approach. Journal of Applied Econometrics, 23(6), 843-860.) Someone is described as "risk averse" if the maximum amount he or she is willing to pay to play is less than $50, the game's expected value. Imagine in this 1, 536-person sample, 1, 482 people are risk averse. Let p denote the proportion of the adult population aged 22 to 55 who are risk averse and 1 - p, the proportion of the same population who are not risk averse. Use the sample results to estimate the proportion p. The proportion 1 p cap of adults in the sample who are risk averse is _________. The proportion 1 - p cap of adults in the sample who are risk averse is __________. You _________ conclude that the sampling distribution of P cap can be approximated by a normal distribution, because ________. The sampling distribution of P cap has a mean _______________ and an estimated standard deviation of _________.

Explanation / Answer

Among 1536 person-sample, 1482 are risk averse, therefore, proportion phat of the adults in the sample who are risk averse is phat=x/n=1482/1536=0.96 [x is number of events and n is number of trials], the proportion 1-phat=1-0.96=0.04 of adults in the sample who are not risk averse.

The sampling distribution phat has a mean 0.96 [mean=phat] and standard deviation 0.005 [sigmap=sqrt{phat(1-phat)/N}=sqrt{0.9691-0.96)/1536}]

The 95% c.i estimate of the proportion of adults aged 22-55 wo are risk averse, p is:phat+-zsqrt[phat(1-phat)/N], where, phat is sample proportion of risk averse, z is z critical score at alpha=0.05 (alpha/2=0.025) and N is sample size.

=phat+-ME=0.96+-1.96*0.005=0.96+-0.0098=(0.9502, 0.9698)

LCL=0.9502 to UCL=0.9698

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