Distribution of the Sample Proportion

Sampling Distribution of a Sample Proportion

The sampling distribution of a sample proportion describes the probability distribution of sample proportions, , obtained from all possible samples of a given size from a population. This concept is fundamental in inferential statistics, as it allows us to make probabilistic statements about sample outcomes.

• Sample Proportion (): The proportion of individuals in a sample with a certain characteristic. It is a random variable because its value varies from sample to sample.

• Population Proportion (): The true proportion of individuals in the population with the characteristic.

• Example: A survey of 1200 adults found that 352 planned to watch the Winter Olympics. The sample proportion is .

Additional info: The sample proportion is used to estimate the population proportion and to conduct hypothesis tests.

Shape, Center, and Spread of the Sampling Distribution

The sampling distribution of has important properties related to its shape, center (mean), and spread (standard deviation).

• Mean of the Sampling Distribution: The mean of the sampling distribution of equals the population proportion .

• Standard Deviation of the Sampling Distribution: The standard deviation (also called the standard error) is given by: where is the sample size.

• Effect of Sample Size:

  • As sample size increases, the standard deviation decreases, making the sampling distribution narrower.

  • As increases, the shape of the sampling distribution becomes more approximately normal (by the Central Limit Theorem), provided certain conditions are met.

• Conditions for Normality: The sampling distribution of is approximately normal if:

• Sample Size Rule: The sample size should be less than 5% of the population size () to ensure independence.

Formulas for the Sampling Distribution of a Sample Proportion

• Mean:

• Standard Deviation:

Examples: Describing and Computing Probabilities for Sample Proportions

• Example 1: Describing the Sampling Distribution Suppose a poll finds that 60% of Americans believe that the state of moral values is declining. In a random sample of 1500 Americans, the sampling distribution of the sample proportion can be described using the formulas above, assuming the normality conditions are met.

• Example 2: Computing Probabilities According to the National Center for Health Statistics, 13% of Americans have hearing trouble.

  • A) In a random sample of 120 Americans, what is the probability that at most 12% have hearing trouble?

  • B) In a random sample of 120 Americans who regularly listen to music using headphones, if 24 have hearing trouble, what might you conclude?

Summary Table: Properties of the Sampling Distribution of a Sample Proportion

Applications

• Estimating population proportions from sample data

• Calculating probabilities for sample outcomes

• Conducting hypothesis tests regarding proportions