According to the Internal Revenue Service, 75 % of all tax...

According to the Internal Revenue Service, $75 \%$ of all tax returns lead to a refund. A random sample of 100 tax returns is taken. a. What is the mean of the distribution of the sample proportion of returns leading to refunds? b. What is the variance of the sample proportion? c. What is the standard error of the sample proportion? d. What is the probability that the sample proportion exceeds 0.8 ?

According to the Internal Revenue Service, $75 \%$ of all tax returns lead to a refund. A random sample of 100 tax returns is taken.
a. What is the mean of the distribution of the sample proportion of returns leading to refunds?
b. What is the variance of the sample proportion?
c. What is the standard error of the sample proportion?
d. What is the probability that the sample proportion exceeds 0.8 ?

Key Concepts

Binomial Distribution

Many problems involving counts of successes in a fixed number of independent trials are modeled by the binomial distribution. This distribution is defined by the number of trials and the probability of success on each trial, and it provides a foundation for understanding outcomes such as the number of refunds in a sample of tax returns.

Sample Proportion

The sample proportion is the ratio of the number of successes to the total number of trials. It is used to estimate the true proportion in a population, and in the binomial setting, it is a transformed version of the binomial variable where each value is scaled by the total number of trials.

Mean of the Distribution

The mean of the sample proportion distribution is derived from the binomial distribution's mean. Specifically, it is equal to the probability of success, reflecting the expected proportion of successes in the sample.

Variance and Standard Error

The variance of the sample proportion is calculated by dividing the variance of the binomial variable by the square of the sample size. The square root of this variance gives the standard error, which quantifies the expected variability of the sample proportion around its mean.

Normal Approximation

For large sample sizes, the distribution of the sample proportion can be approximated using the normal distribution due to the Central Limit Theorem. This approximation simplifies the calculation of probabilities related to the sample proportion, such as determining the probability that it exceeds a given value.

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