Sampling Distributions and Probability of Sample Proportions

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:

$SE = \sqrt{\frac{p(1-p)}{n}}$

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

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

$SE = \sqrt{\frac{0.09 \times 0.91}{1000}} = 0.009$

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:

$z = \frac{\hat{p} - p}{SE}$

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

$z = \frac{0.081 - 0.09}{0.009} = -1$

• 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:

$SE = \sqrt{\frac{0.04 \times 0.96}{300}} = 0.0113$

• Z-score Calculation:

$z = \frac{0.05 - 0.04}{0.0113} = 0.88$

• 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

Step	Description	Formula/Example
1. Identify Parameters	Find population proportion (p), sample size (n), and sample proportion (\( \hat{p} \))	e.g., \( p = 0.09, n = 1000, \hat{p} = 0.081 \)
2. Check Conditions	Ensure \( np \geq 10 \) and \( n(1-p) \geq 10 \)	e.g., \( 1000 \times 0.09 = 90 \)
3. Calculate SE	Compute standard error of \( \hat{p} \)	$SE = \sqrt{\frac{p(1-p)}{n}}$
4. Compute Z-score	Standardize the sample proportion	$z = \frac{\hat{p} - p}{SE}$
5. Find Probability	Use normal table to find area/probability	e.g., Area to left of z = -1 is 0.1611

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.