Final answer:
The mean for the binomial distribution is 40.95. The standard deviation is 5.462. The minimum usual value (u-2) is 29.03 and the maximum usual value (u+2) is 52.92.
Step-by-step explanation:
The mean for a binomial distribution can be calculated using the formula µ = np, where µ is the mean, n is the number of trials, and p is the probability of success. In this case, n is 91 and p is 0.45. Therefore, the mean is µ = 91 * 0.45 = 40.95.
The standard deviation for a binomial distribution can be calculated using the formula σ = √(npq), where σ is the standard deviation and q is 1-p. In this case, n is 91, p is 0.45, and q is 1-0.45 = 0.55. Therefore, the standard deviation is σ = √(91 * 0.45 * 0.55) = 5.462.
The range rule of thumb states that in a normal distribution, the range from the mean minus 2 standard deviations to the mean plus 2 standard deviations includes approximately 95% of the data. Therefore, to find the minimum usual value (u-2), we subtract 2 standard deviations from the mean: u-2 = 40.95 - (2 * 5.462) = 29.03. To find the maximum usual value (u+2), we add 2 standard deviations to the mean: u+2 = 40.95 + (2 * 5.462) = 52.92.