Random samples of size n=500 were selected from a binomial...

Random samples of size $n=500$ were selected from a binomial population with $p=.1 .$ Check to make sure that it is appropriate to use the normal distribution to approximate the sampling distribution of $\hat{p}$. Then use this result to find the probabilities in Exercises $13-15 .$ $$\hat{p}<.10$$

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by the parameters n (the number of trials) and p (the probability of success in each trial), and it forms the basis for understanding discrete probability distributions in many practical scenarios.

Sampling Distribution of the Sample Proportion

When random samples of a fixed size are drawn from a binomial population, the sample proportion (commonly denoted as p?) represents the proportion of successes in the sample. This sampling distribution has a mean equal to p and a variance of p(1-p)/n, reflecting the variability of the estimate across different samples.

Central Limit Theorem for Proportions

The Central Limit Theorem (CLT) guarantees that, for sufficiently large sample sizes, the distribution of the sample proportion will approximately follow a normal distribution even if the underlying data are binomially distributed. This approximation becomes more accurate as the sample size increases.

Conditions for Normal Approximation

To appropriately use the normal distribution as an approximation for the sampling distribution of the sample proportion, it is standard to check that the expected number of successes (np) and failures (n(1-p)) are both greater than or equal to a threshold (commonly 10). This ensures that the binomial distribution is not too skewed and that the CLT applies.

Standard Error Calculation

The standard error of a sample proportion is a measure of its variability across different samples, calculated as the square root of [p(1-p)/n]. This value plays a crucial role in forming confidence intervals and performing hypothesis tests using the normal approximation.

Z-Score Transformation

Once the sampling distribution of the sample proportion is approximated by a normal distribution, the Z-score transformation can be applied. This involves standardizing the sample proportion by subtracting the mean (p) and dividing by the standard error, allowing the use of standard normal tables to compute probabilities for various outcomes.

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