Business Stats problem please help?

The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a 5 year period is .15.

a. Show the sampling distribution of p-bar if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket.

b. What isthe probability that the sample proportion will be within +-.03 of the population proportion?

ALL HELP IS APPRECIATED. If you answer the question, can you PLEASE be specific on how you got the answer.

While it’s not a major problem, it seems to me that p-HAT is the sampling proportion, not p-bar.

Now p-hat will simply be the sample proportion, but to obtain its distribution, it’s useful to go a different route. If you define X1, X2, …, X150, where Xi is equal to 1 if the i-th driver got a ticket and 0 if not, then p-hat = (1/150) * (X1 + X2 + … + X150).

Each Xi is a Bernouilli event with probability 0.15, and the sum will follow a binomial (150, 0.15). So, technically, p-hat would follow a distribution that’s a Binomial (150, 0.15) divided by 150. But that’s very complicated to use. A very good approximation is obtained using the Central Limit Theorem, and we can say that p-hat follows a normal distribution, with mean and variance equal to the Bin/150 distribution.

a. So p-hat (=X-bar by the way) follows a normal with mean=0.15 and standard deviation approximately 0.029155.

b. Then p-hat or X-bar can be treated as any normal variable and it should be easy to figure out its probability of landing within 3% of its mean, since you know the standard deviation.

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