Random samples of size n=75 were selected from a binomial...

Random samples of size $n=75$ were selected from a binomial population with $p=.4 .$ Use the normal distribution to approximate the probabilities given in Exercises 9-12.
$$P(.35 \leq \hat{p} \leq .43)$$

Key Concepts

Z-Score Transformation

Z-score transformation is the process of standardizing a value by subtracting the mean of the distribution and dividing by its standard error. This method allows one to convert the problem into the standard normal framework, where probabilities can be easily obtained from standard normal tables.

Normal Approximation to the Binomial Distribution

When the sample size is large, the binomial distribution, which is inherently discrete, can be approximated by a continuous normal distribution. This approximation simplifies the computation of probabilities for binomial outcomes using z-scores and the normal probability tables.

Central Limit Theorem

The Central Limit Theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean (or proportion) will approximate a normal distribution, even if the underlying population distribution is not normal. This principle is critical in allowing the use of normal approximations in various probability problems.

Sampling Distribution of a Sample Proportion

When sampling from a population, the distribution of the sample proportion p? is approximately normal if the sample size is large enough, with mean equal to the true proportion p and a variance equal to p(1-p)/n. This distribution forms the basis for making probability statements about p?.

Standard Error

The standard error of a sample proportion measures the variability of the sample proportion estimates from sample to sample. It is computed as the square root of [p(1-p)/n] and is essential for quantifying the uncertainty inherent in sample estimates.

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