courses aplia.com 4. Interval estimation of a population proportion Think about

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courses aplia.com 4. Interval estimation of a population proportion Think about the following game: A fair coin is tossed 10 times. Each time the toss results in heads, you receive s10; for tails, you get nothing. What is the maximum amount you would pay to play the game? Define a success as a toss that lands on heads. Then the probability of a success is 0.5, and the expected number of successes after 10 tosses is 10(0,5) 5. Since each success pays s10, the total amount You would expect to win is 5($10) $50. suppose each person in a random sample of 1,536 adults between the ages of 22 and ss 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. Joumal of Applied Econometrics, 23(6), 843-860.) Someone is described as "risk neutral" if the maximum amount he or she is willing to pay to play is equal to $50, the game's expected value. Suppose in this 1,536-person sample, 34 people are risk neutral. Let p denote the proportion of the adult population aged 22 to 55 who are risk neutral and 1 p, the proportion of the same population who are not risk neutral. Use the sample results to estimate the proportion p. The proportion p of adults in the sample who are risk neutral is The proportion 1 p of adults in the sample who are not risk neutral is conclude that the sampling distribution of p can be approximated by a gormal distribution You because The sampling distribution of p has an estimated standard deviation of

Explanation / Answer

proportion phat=34/1536=0.022135

1-phat=0.977865

you can conclude for normal approximation because np>10 and n(1-p)>10

estimated std deviation=(p(1-p)/n)1/2 =0.00375

for 95% CI, z=1.96

hence confidence interval =mean proportion +/- z*std error =0.0148 ; 0.0295

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