Question

suppose that the mean cranial capacity for men is 1180 cc (cubic centimeters) and that the standard deviation is 300 cc. assuming that mens cranial capacities are normally distributed, complete the following statements. (a) approximately 68% of men have cranial capacities between 880 cc and 1480 cc. (b) approximately 99.7% of men have cranial capacities between cc and cc.

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal - distribution: approximately 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$), approximately 95% lies within 2 standard deviations of the mean ($\mu\pm2\sigma$), and approximately 99.7% lies within 3 standard deviations of the mean ($\mu\pm3\sigma$). Given $\mu = 1180$ cc and $\sigma=300$ cc.

Step2: Calculate the range for 68%

For 68% of the data (within 1 standard deviation of the mean), we calculate the lower and upper bounds. The lower bound is $\mu-\sigma=1180 - 300=880$ cc and the upper bound is $\mu+\sigma=1180 + 300 = 1480$ cc.

Step3: Calculate the range for 99.7%

For 99.7% of the data (within 3 standard deviations of the mean), the lower bound is $\mu - 3\sigma=1180-3\times300=1180 - 900 = 280$ cc and the upper bound is $\mu+3\sigma=1180 + 3\times300=1180+900 = 2080$ cc.

Answer:

(a) 1480 cc
(b) 280 cc, 2080 cc