Let's break down the problem step by step for each part:
a) Probability that a person has an IQ greater than 95:
The distribution of IQ scores follows a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 15. We want to find the probability that an individual's IQ is greater than 95.
To find this probability, we calculate the Z-score for an IQ of 95:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Where:
- [tex]\( X \)[/tex] is the IQ score (95 in this case)
- [tex]\( \mu \)[/tex] is the mean (100)
- [tex]\( \sigma \)[/tex] is the standard deviation (15)
Using the Z-score table or a calculator, we find the cumulative probability associated with the calculated Z-score. Since we need the probability of having an IQ greater than 95, we take the complement of this cumulative probability.
The result shows that the probability of an individual having an IQ greater than 95 is 63.1%.
b) Probability that a person has an IQ less than 125:
Similarly, we want to find the probability that an individual's IQ is less than 125. We calculate the Z-score for an IQ of 125:
[tex]\[ Z = \frac{125 - 100}{15} \][/tex]
Using the Z-score, we can find the cumulative probability directly.
The result indicates that the probability of an individual having an IQ less than 125 is 95.2%.
c) Number of people with an IQ less than 110 in a sample of 700 people:
We now want to determine how many people in a sample of 700 will have an IQ less than 110. First, calculate the Z-score for an IQ of 110:
[tex]\[ Z = \frac{110 - 100}{15} \][/tex]
Then, find the corresponding cumulative probability for this Z-score. Multiply this probability by the sample size of 700 to get the number of people.
The result shows that approximately 523 people in the sample would have an IQ less than 110.
d) Number of people with an IQ greater than 140 in a sample of 700 people:
Similarly, calculate the Z-score for an IQ of 140:
[tex]\[ Z = \frac{140 - 100}{15} \][/tex]
Find the cumulative probability for this Z-score and then take its complement to find the probability of having an IQ greater than 140. Multiply this probability by the sample size of 700.
The result indicates that approximately 2 people in the sample would have an IQ greater than 140.
So, summarizing the results:
- Probability of IQ greater than 95: 63.1%
- Probability of IQ less than 125: 95.2%
- Number of people with IQ less than 110 in a sample of 700: 523
- Number of people with IQ greater than 140 in a sample of 700: 2
