Respuesta :
Answer:
b) 84.1%
c) 13.5%
d) 2.35%
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 100
Standard Deviation, σ = 16
We are given that the distribution of IQ tests is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) 68-95-99.7 rule
- Also known as Empirical formula.
- It states that for a normally distributed data , almost all data falls within three standard deviation of mean.
- 68% of data lies within one standard deviation of the mean.
- Thus, 34% of data lies with one standard deviation of mean.
- 95% data falls within two standard deviation of the mean
- 47.5% of data lies with two standard deviation of mean.
- 99.7% of data lies within 3 standard deviation of the mean.
b) percent of people should have IQ scores above 84
P(score greater than 84)
[tex]P( x > 84) = P( z > \displaystyle\frac{84 - 100}{16}) = P(z > -1)[/tex]
[tex]= 1 - P(z \leq -1)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 84) = 1 - 0.159 = 0.841 = 84.1\%[/tex]
84.1% percent of people should have IQ scores above 84.
c) percent of people should have IQ scores between 116 and 132
[tex]116 = 100 + 16 = \mu + \sigma\\132 = 100 + 2(16) = \mu + 2(\sigma)[/tex]
Thus, with the help of empirical rule, we can write,
[tex]P(116<x<132) = 0.475 - 0.34 = 0.135[/tex]
13.5% of people should have IQ scores between 116 and 132.
d) percent of people should have IQ scores below 68
P(IQ scores below 68)
P(x < 68)
[tex]P( x < 68) = P( z < \displaystyle\frac{68 - 100}{16}) = P(z < -2)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 68) = 0.0235 = 2.35\%[/tex]
2.35% of people should have IQ scores below 68.
