Questions: Suppose that the mean cranial capacity for men is 1120 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements. (a) Approximately [] % of men have cranial capacities between 820 cc and 2020 cc. (b) Approximately 95% of men have cranial capacities between [] cc and [] cc.

 Suppose that the mean cranial capacity for men is 1120 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements.

(a) Approximately [] % of men have cranial capacities between 820 cc and 2020 cc.

(b) Approximately 95% of men have cranial capacities between [] cc and [] cc.
Transcript text: Suppose that the mean cranial capacity for men is 1120 cc (cubic centimeters) and that the standard deviation is 300 cc. Assuming that men's cranial capacities are normally distributed, complete the following statements. (a) Approximately [] % of men have cranial capacities between 820 cc and 2020 cc. (b) Approximately 95% of men have cranial capacities between [] cc and [] cc.
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Solution

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Solution Steps

Step 1: Calculate the Percentage of Men with Cranial Capacities Between 820 cc and 2020 cc

To find the percentage of men with cranial capacities between 820cc 820 \, \text{cc} and 2020cc 2020 \, \text{cc} , we first calculate the cumulative distribution function (CDF) values at these points:

  • P(X820)0.1587 P(X \leq 820) \approx 0.1587
  • P(X2020)0.9987 P(X \leq 2020) \approx 0.9987

The percentage of men with cranial capacities in this range is given by:

P(820<X<2020)=P(X2020)P(X820)0.99870.1587=0.84 P(820 < X < 2020) = P(X \leq 2020) - P(X \leq 820) \approx 0.9987 - 0.1587 = 0.84

Thus, approximately 84.00% 84.00\% of men have cranial capacities between 820cc 820 \, \text{cc} and 2020cc 2020 \, \text{cc} .

Step 2: Calculate the Cranial Capacities for 95% of Men

To determine the cranial capacities that encompass approximately 95% 95\% of men, we first find the Z critical value for a 95% 95\% confidence level:

Z=1.96 Z = 1.96

Using this Z value, we can calculate the lower and upper bounds of cranial capacities:

Lower Bound=μZσ=11201.96300532.00cc \text{Lower Bound} = \mu - Z \cdot \sigma = 1120 - 1.96 \cdot 300 \approx 532.00 \, \text{cc} Upper Bound=μ+Zσ=1120+1.963001708.00cc \text{Upper Bound} = \mu + Z \cdot \sigma = 1120 + 1.96 \cdot 300 \approx 1708.00 \, \text{cc}

Thus, approximately 95% 95\% of men have cranial capacities between 532.00cc 532.00 \, \text{cc} and 1708.00cc 1708.00 \, \text{cc} .

Final Answer

  • Approximately 84.00% 84.00\% of men have cranial capacities between 820cc 820 \, \text{cc} and 2020cc 2020 \, \text{cc} .
  • Approximately 95% 95\% of men have cranial capacities between 532.00cc 532.00 \, \text{cc} and 1708.00cc 1708.00 \, \text{cc} .

84.00% and (532.00cc,1708.00cc) \boxed{84.00\% \text{ and } (532.00 \, \text{cc}, 1708.00 \, \text{cc})}

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