To find the percentage of men with cranial capacities between 820 cc 820 \, \text{cc} 820cc and 2020 cc 2020 \, \text{cc} 2020cc, we first calculate the cumulative distribution function (CDF) values at these points:
The percentage of men with cranial capacities in this range is given by:
P(820<X<2020)=P(X≤2020)−P(X≤820)≈0.9987−0.1587=0.84 P(820 < X < 2020) = P(X \leq 2020) - P(X \leq 820) \approx 0.9987 - 0.1587 = 0.84 P(820<X<2020)=P(X≤2020)−P(X≤820)≈0.9987−0.1587=0.84
Thus, approximately 84.00% 84.00\% 84.00% of men have cranial capacities between 820 cc 820 \, \text{cc} 820cc and 2020 cc 2020 \, \text{cc} 2020cc.
To determine the cranial capacities that encompass approximately 95% 95\% 95% of men, we first find the Z critical value for a 95% 95\% 95% confidence level:
Z=1.96 Z = 1.96 Z=1.96
Using this Z value, we can calculate the lower and upper bounds of cranial capacities:
Lower Bound=μ−Z⋅σ=1120−1.96⋅300≈532.00 cc \text{Lower Bound} = \mu - Z \cdot \sigma = 1120 - 1.96 \cdot 300 \approx 532.00 \, \text{cc} Lower Bound=μ−Z⋅σ=1120−1.96⋅300≈532.00cc Upper Bound=μ+Z⋅σ=1120+1.96⋅300≈1708.00 cc \text{Upper Bound} = \mu + Z \cdot \sigma = 1120 + 1.96 \cdot 300 \approx 1708.00 \, \text{cc} Upper Bound=μ+Z⋅σ=1120+1.96⋅300≈1708.00cc
Thus, approximately 95% 95\% 95% of men have cranial capacities between 532.00 cc 532.00 \, \text{cc} 532.00cc and 1708.00 cc 1708.00 \, \text{cc} 1708.00cc.
84.00% and (532.00 cc,1708.00 cc) \boxed{84.00\% \text{ and } (532.00 \, \text{cc}, 1708.00 \, \text{cc})} 84.00% and (532.00cc,1708.00cc)
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