91Ó°ÊÓ

In statistics, the normal distribution is a continuous probability distribution that is symmetrical around its mean. It is often referred to as the "bell curve" due to its distinctive shape. The normal distribution is crucial in statistical inference because it allows for approximation and modeling of real-world variables that tend to cluster around a central value.

One key property of the normal distribution is the empirical rule, which states that:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% falls within two standard deviations.
• Approximately 99.7% falls within three standard deviations.

The relevance of this concept in the given problem arises when deciding if a binomial distribution can be approximated by a normal distribution. The rule of thumb is that a normal approximation is suitable if both \(np\) and \(n(1-p)\) are greater than 5, ensuring enough observations for the approximation to hold true. Given \(np = 30\) and \(n(1-p) = 70\), both conditions are satisfied, allowing us to proceed with the normal approximation for the calculation of confidence intervals.

A confidence interval provides a range of values that are likely to contain a population parameter, in this case, the true population proportion \(p\). The interval is defined by two limits—the lower and upper bounds—calculated from the sample data and built to contain the parameter of interest with a certain confidence level.

In the problem, a 90% confidence interval means that if we were to take many samples and calculate the interval for each, then approximately 90% of these intervals would contain the population proportion \(p\). It's a way of expressing uncertainty and ensuring that our statistical estimates reflect reality.

This is computed using the formula:\[\hat{p} \pm z \times SE(\hat{p})\]Here, \(\hat{p}\) is the sample proportion, \(z\) is the critical value from the standard normal distribution for the desired confidence level (1.645 for 90%), and \(SE(\hat{p})\) is the standard error of the sample proportion. The result of our calculation yields the interval \((0.2246, 0.3754)\), indicating where the true population proportion is likely located with 90% confidence.

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. Each trial has two possible outcomes: success or failure. This distribution is defined by two parameters:

• \(n\), the number of trials
• \(p\), the probability of success on an individual trial

In the exercise, the distribution is characterized by 100 trials and a success probability, which was estimated from the sample to be 0.3.

The binomial distribution is important for understanding the likelihood of different outcomes over a series of trials. When the number of trials is sufficiently large and the success probability is not too extreme (neither too close to 0 nor 1), the binomial distribution can be approximated by a normal distribution. This is useful as it simplifies the computation of probabilities and confidence intervals, allowing us to apply tools from the continuous realm, like the normal distribution, to our discrete data.