An administrator for a large group of hospitals believes...

An administrator for a large group of hospitals believes that of all patients $30 \%$ will generate bills that become at least 2 months overdue. A random sample of 200 patients is taken. a. What is the standard error of the sample proportion that will generate bills that become at least 2 months overdue? b. What is the probability that the sample proportion is less than 0.25 ? c. What is the probability that the sample proportion is more than 0.33 ? d. What is the probability that the sample proportion is between 0.27 and 0.33 ?

An administrator for a large group of hospitals believes that of all patients $30 \%$ will generate bills that become at least 2 months overdue. A random sample of 200 patients is taken.
a. What is the standard error of the sample proportion that will generate bills that become at least 2 months overdue?
b. What is the probability that the sample proportion is less than 0.25 ?
c. What is the probability that the sample proportion is more than 0.33 ?
d. What is the probability that the sample proportion is between 0.27 and 0.33 ?

Key Concepts

Z-Scores and Probability Calculation

A Z-score represents how many standard deviations a point is from the mean of a normally distributed variable. In the context of the sample proportion, converting the sample proportion to a Z-score using the formula Z = (observed proportion - true proportion) / standard error enables the use of the standard normal cumulative distribution function to determine the probability of observing a sample proportion less than, greater than, or between given values.

Normal Approximation to the Binomial

When dealing with proportions from binomial distributions, if the sample size is large and the product of the sample size with the probabilities (np and n(1 - p)) are both sufficiently large, it allows us to approximate the binomial distribution with a normal distribution. This approximation facilitates the calculation of probabilities related to the sample proportion using Z-scores.

Central Limit Theorem

The central limit theorem states that, for a sufficiently large sample size, the sampling distribution of the sample proportion tends to be approximately normally distributed, regardless of the distribution of the underlying population. This is a crucial justification for using normal probability methods to calculate probabilities concerning the sample proportion when the sample size is large.

Standard Error of a Sample Proportion

This concept refers to the measure of variability or dispersion of the sample proportion from the true population proportion. It is calculated using the formula ?[p(1 - p) / n], where p is the true population proportion and n is the sample size. It quantifies the expected spread in the sample proportions if different samples were taken from the same population.

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