A random sample of n=250 measurements is drawn from NW a...

A random sample of $n=250$ measurements is drawn from NW a binomial population with probability of success 85 .
a. Find $E(\hat{p})$ and $\sigma_{\hat{p}}$.
b. Describe the shape of the sampling distribution of $\hat{p}$.
c. Find $P(\hat{p}<9)$.

Key Concepts

Sample Proportion

The sample proportion is a statistic that estimates the probability of a success in a binomial experiment. It is calculated by dividing the number of successes in the sample by the total number of trials, and it serves as an unbiased estimator for the true probability of success.

Expected Value and Standard Error of the Proportion

The expected value of the sample proportion is equal to the success probability of the underlying population. The standard error, which measures the variability of the sample proportion, is given by the square root of the product of the success probability and the failure probability divided by the sample size. This quantifies how much the sample proportion is likely to fluctuate from sample to sample.

Central Limit Theorem

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample proportion will be approximately normally distributed regardless of the shape of the original distribution. This is particularly useful in statistics when dealing with binomial data as it allows for the use of normal probability methods for inference.

Normal Approximation for Proportions

When the sample size is large, the binomial distribution of the sample proportion can be approximated by a normal distribution. This approximation simplifies the calculation of probabilities associated with the sample proportion, facilitating the use of standard normal distribution tables and z-scores.

Using Z-scores for Probability Calculations

Z-scores standardize the distance of a sample statistic from its mean in terms of standard deviations. Converting the sample proportion to a z-score allows for the computation of probabilities using the standard normal distribution, making it easier to answer questions about the likelihood of observing certain sample proportions.

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