Sampling Distributions of Sample Proportions

Introduction

Sampling distributions are fundamental in statistics for understanding how sample statistics (like proportions) behave when samples are repeatedly drawn from a population. This section focuses on the distribution of sample proportions, standard error calculations, and probability estimations using the normal approximation.

Sample Proportion and Expected Value

• Sample Proportion (\( \hat{p} \)): The proportion of items in a sample with a certain characteristic (e.g., the proportion of 't's in a sample of letters).

• Expected Value: The mean of the sampling distribution of \( \hat{p} \) is equal to the population proportion \( p \).

• Example: If the population proportion of 't's is 9% (\( p = 0.09 \)), then in a random sample of 1000 letters, the expected sample proportion is also 9%.

Standard Error of the Sample Proportion

• Definition: The standard error (SE) measures the variability of the sample proportion from sample to sample.

• Formula:

• Where \( p \) is the population proportion and \( n \) is the sample size.

• Example Calculation: For \( p = 0.09 \), \( n = 1000 \):

Using the Normal Approximation

• When the sample size is large and both \( np \) and \( n(1-p) \) are greater than 10, the sampling distribution of \( \hat{p} \) can be approximated by a normal distribution.

• Standardization (Z-score): To find probabilities, convert the sample proportion to a z-score:

• Example: To find the probability that the sample proportion is 8.1% or less (\( \hat{p} \leq 0.081 \)):

• Using the standard normal table, the area to the left of \( z = -1 \) is approximately 0.1611. This is the probability that the sample proportion is 8.1% or less.

Interpreting Probability Results

• Probability Statement: "The probability that the random sample of 1000 letters will contain 8.1% or fewer 't's is 0.1611."

• Application: Such calculations are useful for quality control, polling, and other inferential statistics applications.

Probability of Sample Proportions in Context

Example: Proportion of Vegans in a Restaurant

Suppose 4% of the population are vegans. In a random sample of 300 patrons, what is the probability that 15 or more are vegans?

• Population Proportion: \( p = 0.04 \)

• Sample Size: \( n = 300 \)

• Sample Proportion Threshold: \( \hat{p} = \frac{15}{300} = 0.05 \)

• Standard Error:

• Z-score Calculation:

• Using the standard normal table, the area to the right of \( z = 0.88 \) is approximately 0.1894.

• Interpretation: There is about an 18.94% probability that 15 or more patrons will be vegan, which may indicate the restaurant could run out of vegan meals.

Conditions for Normal Approximation

• Check that \( np \geq 10 \) and \( n(1-p) \geq 10 \) before using the normal approximation.

• For the vegan example: \( 300 \times 0.04 = 12 \) and \( 300 \times 0.96 = 288 \), both satisfy the condition.

Summary Table: Steps for Calculating Probability of Sample Proportions

Additional info:

• These methods are widely used in hypothesis testing and confidence interval estimation for proportions.

• Always check the normal approximation conditions before applying the z-score method.