Sampling Distribution of Sample Proportion

Introduction to Sampling Distributions

The sampling distribution of a sample proportion describes the distribution of proportions obtained from repeated random samples of a fixed size from a population. This concept is fundamental in inferential statistics, as it allows us to estimate probabilities and make inferences about population parameters.

• Sample Proportion (p̂): The proportion of successes in a sample, calculated as , where x is the number of successes and n is the sample size.

• Binomial Distribution: Used when each trial has two possible outcomes (success/failure) and the probability of success is constant.

• Normal Approximation: For large sample sizes, the binomial distribution can be approximated by a normal distribution if and .

Example: If you flip a coin 10 times and count the number of heads, the distribution of the number of heads is binomial. If you take many samples and calculate the proportion of heads in each, you get the sampling distribution of the sample proportion.

Formulas for Binomial and Normal Approximations

When the conditions for normal approximation are met, probabilities for binomial outcomes can be estimated using the normal distribution.

• Binomial Probability Formula:

• Mean and Standard Deviation of Sample Proportion: Mean: Standard deviation:

• Normal Approximation: Convert the binomial variable to a z-score:

Example: To find the probability that between 62 and 70 people out of 100 vote for a candidate (with p = 0.48), use the normal approximation to estimate .

Steps for Using Normal Approximation

Follow these steps to approximate binomial probabilities using the normal distribution:

1. Verify that and .

2. Calculate the mean and standard deviation for the sample proportion.

3. Apply the continuity correction (add or subtract 0.5 to the endpoints when converting discrete binomial to continuous normal).

4. Convert the values to z-scores using the formula above.

5. Use the standard normal table or calculator to find the required probability.

Continuity Correction

When approximating a binomial distribution with a normal distribution, a continuity correction is applied to account for the fact that the binomial is discrete and the normal is continuous. This involves adjusting the endpoints by 0.5.

• For , use

• For , use

• For , use

Summary Table: Continuity Correction Applications

Applications and Examples

Example 1: Voting Proportion

Suppose the probability of someone voting for a candidate is 48%. To approximate the probability that between 62 and 70 people out of a sample of 100 vote for the candidate, use the normal approximation as described above.

Example 2: Product Preference

If a study finds that 80% of people prefer Pepsi over Coca Cola, use the normal approximation to estimate the probability that between 10 and 11 people out of a sample of 100 prefer Coca Cola.

Example 3: Car Recalls

In 2023, about 36 million cars were recalled out of 94 million produced. For a dealer with 76 cars, use the normal approximation to estimate the probability that more than half were recalled.

Key Terms and Definitions

• Binomial Variable (X): Number of successes in n independent trials.

• Sample Proportion (p̂): Proportion of successes in a sample.

• Sampling Distribution: Distribution of a statistic (like p̂) over repeated samples.

• Normal Approximation: Using the normal distribution to estimate binomial probabilities when sample size is large.

• Continuity Correction: Adjustment made when using a continuous distribution to approximate a discrete one.

Comparison: Binomial vs. Normal Approximation

Additional info: These notes expand on the provided slides by including definitions, formulas, and step-by-step procedures for using the normal approximation to estimate binomial probabilities, as well as practical examples and comparison tables for clarity.