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 $13-15 .$ $$\hat{p}>.12$$

Key Concepts

Conditions for Normal Approximation

Before applying the normal approximation to the binomial (or the sampling distribution of a proportion), it is necessary to verify that the expected numbers of successes and failures (np and n(1-p)) are both greater than a designated threshold (commonly 5). This check ensures that the binomial distribution is not too skewed and that the normal approximation will be reasonably accurate.

Sampling Distribution of a Sample Proportion

This concept describes the distribution of the proportion of successes in repeated samples drawn from a population. It is the foundation for making inferences about the true proportion in the population and is characterized by its mean (which equals the population proportion) and its standard deviation, which depends on the sample size and the population proportion.

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the distribution of the sample means (or proportions) will approximate a normal distribution regardless of the shape of the population distribution. This theorem is crucial because it justifies the use of the normal model when analyzing sample proportions from a binomial population.

Normal Approximation to the Binomial Distribution

This concept involves approximating the binomial distribution with a normal distribution when certain conditions are met. The approximation works well when the sample size is large and the probability of success is not too close to 0 or 1, enabling simpler probability calculations.

Z-Score Standardization

This technique involves converting a random variable into a standard normal variable (z-score) by subtracting the mean and dividing by the standard deviation. Standardization allows for the utilization of the standard normal distribution to compute probabilities, such as determining the likelihood that a sample proportion exceeds a given value.

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