Question

1. -/3 points according to the empirical rule, approximately what percentage of the area under a normal distribution lies within 1 standard deviation? % within 2 standard deviations? % within 3 standard deviations? % submit answer 3. -/1 points can a normal distribution always be used to approximate a binomial distribution? explain your answer. yes. we can always use the normal distribution to approximate the binomial distribution. no. for the approximation to be accurate, np > 10 and nq > 10. no. for the approximation to be accurate, np > 5 and nq > 5. no. for the approximation to be accurate, np > 5 and nq < 5.

Explanation:

Step-by-Step Format:

Step1: Recall the empirical rule

The empirical rule (68 - 95 - 99.7 rule) for a normal distribution states that:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\)), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\)).
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\)).

The normal distribution can be used to approximate the binomial distribution \(B(n,p)\) when \(np>5\) and \(nq > 5\) (where \(q=1 - p\)). This is because when \(np\) and \(nq\) are large enough, the binomial distribution becomes symmetric and bell - shaped (similar to the normal distribution). If \(np\leq5\) or \(nq\leq5\), the binomial distribution is too skewed for a good normal approximation.

Answer:

Within 1 standard deviation: \(68\%\)
Within 2 standard deviations: \(95\%\)
Within 3 standard deviations: \(99.7\%\)

For the second question (about normal - binomial approximation):