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Question 1170955: 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
P
(IQ greater than 95) = %
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
P
(IQ less than 125) = %
c) In a sample of 600 people, how many people would have an IQ less than 110?
people
d) In a sample of 600 people, how many people would have an IQ greater than 140?
people
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve each part of the problem:
**Given:**
* Mean (μ) = 100
* Standard deviation (σ) = 15
* IQ is normally distributed.
**a) Probability of IQ greater than 95:**
1. **Calculate the z-score:**
* z = (x - μ) / σ
* z = (95 - 100) / 15 = -5 / 15 = -1/3 ≈ -0.3333
2. **Find the probability:**
* Use a z-table or calculator to find P(Z > -0.3333).
* P(Z > -0.3333) = 1 - P(Z < -0.3333) ≈ 1 - 0.3707 ≈ 0.6293
3. **Convert to percentage and round:**
* 0.6293 * 100 = 62.93%
* Rounded to the nearest tenth: 62.9%
**b) Probability of IQ less than 125:**
1. **Calculate the z-score:**
* z = (125 - 100) / 15 = 25 / 15 = 5/3 ≈ 1.6667
2. **Find the probability:**
* Use a z-table or calculator to find P(Z < 1.6667).
* P(Z < 1.6667) ≈ 0.9525
3. **Convert to percentage and round:**
* 0.9525 * 100 = 95.25%
* Rounded to the nearest tenth: 95.3%
**c) Number of people with IQ less than 110 in a sample of 600:**
1. **Calculate the z-score:**
* z = (110 - 100) / 15 = 10 / 15 = 2/3 ≈ 0.6667
2. **Find the probability:**
* Use a z-table or calculator to find P(Z < 0.6667).
* P(Z < 0.6667) ≈ 0.7475
3. **Calculate the expected number:**
* Expected number = 600 * 0.7475 ≈ 448.5
* Rounded to the nearest whole number: 449
**d) Number of people with IQ greater than 140 in a sample of 600:**
1. **Calculate the z-score:**
* z = (140 - 100) / 15 = 40 / 15 = 8/3 ≈ 2.6667
2. **Find the probability:**
* Use a z-table or calculator to find P(Z > 2.6667).
* P(Z > 2.6667) = 1 - P(Z < 2.6667) ≈ 1 - 0.9962 ≈ 0.0038
3. **Calculate the expected number:**
* Expected number = 600 * 0.0038 ≈ 2.28
* Rounded to the nearest whole number: 2
**Answers:**
* a) P(IQ greater than 95) = 62.9%
* b) P(IQ less than 125) = 95.3%
* c) 449 people
* d) 2 people
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