The proportion of individuals insured by the All-Driver...

The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is .15 . a. Show the sampling distribution of $\bar{p}$ if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket. b. What is the probability that the sample proportion will be within ±.03 of the population proportion?

The proportion of individuals insured by the All-Driver Automobile Insurance Company who received at least one traffic ticket during a five-year period is .15 .
a. Show the sampling distribution of $\bar{p}$ if a random sample of 150 insured individuals is used to estimate the proportion having received at least one ticket.
b. What is the probability that the sample proportion will be within ±.03 of the population proportion?

Key Concepts

Probability Calculations Using the Normal Distribution

This concept involves finding the likelihood that a sample statistic falls within a specified interval around the true population parameter. When the sampling distribution is approximately normal, the interval can be converted into z-scores using the standard error, and standard normal probability tables or calculators can then be used to determine probabilities associated with these z-scores.

Central Limit Theorem

The Central Limit Theorem explains that given a sufficiently large sample size, the distribution of the sample proportion will approximate a normal distribution regardless of the shape of the population distribution. This allows for the use of z-scores and normal probability calculations even when dealing with proportions as long as the conditions (np and n(1-p) both greater than or equal to 10) are met.

Sampling Distribution of the Sample Proportion

This concept refers to the probability distribution of sample proportions, which represents the variation in the proportion of successes obtained from different samples of the same size drawn from a population. In cases where the sample size is sufficiently large, the distribution is approximately normally distributed with a mean equal to the true population proportion and a standard deviation (standard error) that depends on the population proportion and the sample size.

Standard Error of the Sample Proportion

The standard error is the standard deviation of the sampling distribution of the sample proportion. It quantifies how much the sample proportion is expected to fluctuate around the true population proportion from one sample to another. The formula for the standard error is given by the square root of [p(1-p)/n], where p is the true population proportion and n is the sample size.

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