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Chapter 6: Problem 103

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of \(.02\). According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

Short Answer

Expert verified
The mean and standard deviation of this binomial distribution is 0.3 and 0.77 respectively. Using the empirical rule (68-95-99.7), we find that 68% of the data falls within the range of -0.47 to 1.07, which includes negative numbers that are unacceptable for the number of successes or trials in binomial distribution. This deviation into non-sensical negative numbers indicates the imprudent application of the normal distribution approximation for this binomial distribution.

Step by step solution

01

Calculate the Mean and Standard Deviation

Firstly, let's calculate the mean and standard deviation of the binomial distribution. The mean \(\mu\) of a binomial distribution is given by the formula \(\mu = np\) where \(n\) is the number of trials and \(p\) is the probability of success. Here, \(n = 15\) and \(p = .02\), so \(\mu = 15 * .02 = 0.3\).\n\nThe standard deviation \(\sigma\) is given by the formula \(\sigma = \sqrt{np(1-p)}\). Substituting our values, we find \(\sigma = \sqrt{15*.02*(1-.02)} = 0.77\) (rounded to two decimal places).
02

Apply the Empirical Rule

Next, let's consider the empirical rule, which is oftentimes referred to as the 68-95-99.7 rule. This means that 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.\n\nWith a mean of 0.3 and a standard deviation of 0.77, one standard deviation from the mean (0.3 \(\pm\) 0.77) would give us a range from -0.47 to 1.07. However, a negative number of successes or trials is unacceptable in the binomial distribution.
03

Interpret the Results

According to the empirical rule, 68% of the data should fall within -0.47 to 1.07. However, negative numbers are non-sensical in this context and suggest that applying the normal approximation to the binomial distribution would be inappropriate in this situation, even if the sample size was appropriate. The nonzero probability of unacceptable (negative) outcomes illustrates the issue.

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Most popular questions from this chapter

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