Questions: Chapter 6 Homework - Normal Distribution Score: 10.75/15 Answered: 11/15 Question 11 Score on last try: 0.75 of 1 pts. See Details for more. IQ is normally distributed with a mean of 100 and a standard deviation of 15. a) Suppose one individual is randomly chosen. Find the probability that this person has an IQ greater than 95. Write your answer in percent form. Round to the nearest tenth of a percent. P(IQ greater than 95)=62.9% b) Suppose one individual is randomly chosen. Find the probability that this person has an IQ less than 125. Write your answer in percent form. Round to the nearest tenth of a percent. P(IQ less than 125)=95.3% c) In a sample of 800 people, how many people would have an IQ less than 110? 374 people d) In a sample of 800 people, how many people would have an IQ greater than 140? 8 people

Chapter 6 Homework - Normal Distribution Score: 10.75/15 Answered: 11/15 Question 11 Score on last try: 0.75 of 1 pts. See Details for more. IQ is normally distributed with a mean of 100 and a standard deviation of 15. a) Suppose one individual is randomly chosen. Find the probability that this person has an IQ greater than 95. Write your answer in percent form. Round to the nearest tenth of a percent. P(IQ greater than 95)=62.9% b) Suppose one individual is randomly chosen. Find the probability that this person has an IQ less than 125. Write your answer in percent form. Round to the nearest tenth of a percent. P(IQ less than 125)=95.3% c) In a sample of 800 people, how many people would have an IQ less than 110? 374 people d) In a sample of 800 people, how many people would have an IQ greater than 140? 8 people
Transcript text: Chapter 6 Homework - Normal Distribution Score: 10.75/15 Answered: 11/15 Question 11 Score on last try: $\mathbf{0 . 7 5}$ of $\mathbf{1}$ pts. See Details for more. IQ is normally distributed with a mean of 100 and a standard deviation of 15. a) Suppose one individual is randomly chosen. Find the probability that this person has an IQ greater than 95. Write your answer in percent form. Round to the nearest tenth of a percent. \[ P(\text { IQ greater than } 95)=62.9 \% \] b) Suppose one individual is randomly chosen. Find the probability that this person has an IQ less than 125. Write your answer in percent form. Round to the nearest tenth of a percent. \[ P(\text { IQ less than } 125)=95.3 \% \] c) In a sample of 800 people, how many people would have an IQ less than 110? \[ 374 \quad \text { people } \] d) In a sample of 800 people, how many people would have an IQ greater than 140? \[ 8 \quad \text { people } \]
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Solution

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Solution Steps

Step 1: Probability that an individual has an IQ greater than 95

To find the probability that an individual has an IQ greater than 95, we calculate:

\[ P(\text{IQ} > 95) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-0.3333) = 0.6306 \]

Converting this probability to percentage form:

\[ P(\text{IQ} > 95) = 0.6306 \times 100 = 63.1\% \]

Step 2: Probability that an individual has an IQ less than 125

Next, we calculate the probability that an individual has an IQ less than 125:

\[ P(\text{IQ} < 125) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.6667) - \Phi(-\infty) = 0.9522 \]

Converting this probability to percentage form:

\[ P(\text{IQ} < 125) = 0.9522 \times 100 = 95.2\% \]

Step 3: Number of people with an IQ less than 110 in a sample of 800

To find the number of people with an IQ less than 110 in a sample of 800, we calculate:

\[ P(\text{IQ} < 110) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.6667) - \Phi(-\infty) = 0.7475 \]

The expected number of people is:

\[ \text{Number of people} = P(\text{IQ} < 110) \times 800 = 0.7475 \times 800 = 598 \]

Final Answer

The results are as follows:

  • Probability that an individual has an IQ greater than 95: \( \boxed{63.1\%} \)
  • Probability that an individual has an IQ less than 125: \( \boxed{95.2\%} \)
  • Number of people with an IQ less than 110 in a sample of 800: \( \boxed{598} \)
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