Section 8.2

Distribution of the Sample Proportion

Understand the sampling distribution of proportions and calculate probabilities using the normal approximation.

Sample Proportion

The sample proportion is the fraction of individuals in a sample with a specific characteristic.

x = number of individuals with the characteristic

n = sample size

Mean of Sampling Distribution

Equals the population proportion

Standard Deviation

Decreases as n increases

The sampling distribution of is approximately normal if both conditions are met:

Ensures enough successes & failures for normal shape

Sample is less than 5% of population

To find probabilities about , transform it into a Z-score:

After calculating Z: Use the standard normal table (Table V) or technology to find the corresponding probability.

Population Proportion (p)

Sample Size (n)

Sample Proportion (p̂)

Probability Type

Normal Approximation Valid

np(1-p) = 21.00 ≥ 10 ✓

:0.3000

:0.0458

Z-score:-1.0911

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Probability

0.137617

P(p̂ < 0.25)

Z-Score Calculation

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