Respuesta :
Answer:
a) 0.1587 b) 0.023 c) 0.1587 d) 1.15 e)-0.95
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 510
Standard Deviation, σ = 100
We are given that the distribution of SAT score is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) P(score greater than 610)
P(x > 610)
[tex]P( x > 610) = P( z > \displaystyle\frac{610 - 510}{100}) = P(z > 1)[/tex]
[tex]= 1 - P(z \leq 1)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 610) = 1 - 0.8413 = 0.1587 = 15.87\%[/tex]
b) P(score greater than 710)
[tex]P(x > 710) = P(z > \displaystyle\frac{710-510}{100}) = P(z > 2)\\\\P( z > 2) = 1 - P(z \leq 2)[/tex]
Calculating the value from the standard normal table we have,
[tex]1 - 0.977 = 0.023 = 2.3\%\\P( x > 710) = 2.3\%[/tex]
c)P(score between 410 and 510)
[tex]P(410 \leq x \leq 510) = P(\displaystyle\frac{410 - 510}{100} \leq z \leq \displaystyle\frac{510-510}{100}) = P(-1 \leq z \leq 0)\\\\= P(z \leq 0) - P(z < -1)\\= 0.500 - 0.159 = 0.341 = 34.1\%[/tex]
[tex]P(410 \leq x \leq 510) = 34.1\%[/tex]
d) x = 625
[tex]z_{score} = \displaystyle\frac{625 - 510}{100} = \displaystyle\frac{115}{100} = 1.15[/tex]
e) x = 415
[tex]z_{score} = \displaystyle\frac{415 - 510}{100} = \displaystyle\frac{-95}{100} = -0.95[/tex]
