To find the probability that an individual has an IQ greater than 95, we calculate:
P(IQ>95)=Φ(Zend)−Φ(Zstart)=Φ(∞)−Φ(−0.3333)=0.6306 P(\text{IQ} > 95) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-0.3333) = 0.6306 P(IQ>95)=Φ(Zend)−Φ(Zstart)=Φ(∞)−Φ(−0.3333)=0.6306
Converting this probability to percentage form:
P(IQ>95)=0.6306×100=63.1% P(\text{IQ} > 95) = 0.6306 \times 100 = 63.1\% P(IQ>95)=0.6306×100=63.1%
Next, we calculate the probability that an individual has an IQ less than 125:
P(IQ<125)=Φ(Zend)−Φ(Zstart)=Φ(1.6667)−Φ(−∞)=0.9522 P(\text{IQ} < 125) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.6667) - \Phi(-\infty) = 0.9522 P(IQ<125)=Φ(Zend)−Φ(Zstart)=Φ(1.6667)−Φ(−∞)=0.9522
P(IQ<125)=0.9522×100=95.2% P(\text{IQ} < 125) = 0.9522 \times 100 = 95.2\% P(IQ<125)=0.9522×100=95.2%
To find the number of people with an IQ less than 110 in a sample of 800, we calculate:
P(IQ<110)=Φ(Zend)−Φ(Zstart)=Φ(0.6667)−Φ(−∞)=0.7475 P(\text{IQ} < 110) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.6667) - \Phi(-\infty) = 0.7475 P(IQ<110)=Φ(Zend)−Φ(Zstart)=Φ(0.6667)−Φ(−∞)=0.7475
The expected number of people is:
Number of people=P(IQ<110)×800=0.7475×800=598 \text{Number of people} = P(\text{IQ} < 110) \times 800 = 0.7475 \times 800 = 598 Number of people=P(IQ<110)×800=0.7475×800=598
The results are as follows:
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