© BFW Publishers Suppose Kristen is researching failures in the restaurant business. In the city where she lives, the probability that an independent restaurant will fail in the first year is \( 51 \% \). She obtains a random sample of 84 independent restaurants that opened in her city more than one year ago and determines if each one had closed within a year. What are the mean and standard deviation of the number of restaurants that failed within a year? Please give your answers precise to two decimal places. \[ \mu=\square \quad \sigma=\square \]

To find the mean (\(\mu\)) and the standard deviation (\(\sigma\)) of the number of restaurants that failed within a year, we can use the properties of a binomial distribution since each restaurant either fails or does not with a fixed probability. 1. **Mean (\(\mu\))**: The mean of a binomial distribution can be calculated using the formula: \[ \mu = n \cdot p \] where \(n\) is the number of trials (restaurants) and \(p\) is the probability of success (failure in this context). Given: - \(n = 84\) - \(p = 0.51\) So, \[ \mu = 84 \cdot 0.51 = 42.84 \] 2. **Standard Deviation (\(\sigma\))**: The standard deviation of a binomial distribution can be calculated using the formula: \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \] Here, \(1 - p = 0.49\). Therefore, \[ \sigma = \sqrt{84 \cdot 0.51 \cdot 0.49} \] \[ = \sqrt{84 \cdot 0.2499} \approx \sqrt{21} \approx 4.58 \] Thus, the answers are: \[ \mu = 42.84 \quad \sigma = 4.58 \]