Complete Question
The complete question is shown on the first uploaded image
Answer:
a
[tex]P(X < 4) = 0.4141 [/tex]
b
[tex]P(4 < X < 6) = 0.32876[/tex]
c
[tex]P(X >8) = 0.064037 [/tex]
d
The correct option is B
Step-by-step explanation:
From the question we are told that
The mean is [tex]\mu = 4.5[/tex]
The standard deviation is [tex]\sigma = 2.3[/tex]
Generally the probability that a randomly selected study participant's response was less than 4 is mathematically represented as
[tex]P(X < 4) = P(\frac{X - \mu }{\sigma} < \frac{4 - 4.5}{ 2.3} )[/tex]
[tex]\frac{X -\mu}{\sigma} = Z (The \ standardized \ value \ of\ X )[/tex]
So
[tex]P(X < 4) = P(Z < -0.217 )[/tex]
From the z-table [tex]P(Z < -0.217 ) = 0.4141[/tex]
So
[tex]P(X < 4) = 0.4141 [/tex]
Generally the probability that a randomly selected study participant's response was between 4 and 6 is mathematically represented as
[tex]P(4 < X < 6) = P( < \frac{4 - 4.5}{ 2.3} < \frac{X - \mu }{\sigma} < \frac{6 - 4.5}{2.3})[/tex]
=> [tex]P(4 < X < 6) = P( \frac{4 - 4.5}{ 2.3} < \frac{X - \mu }{\sigma} < \frac{6 - 4.5}{2.3})[/tex]
=> [tex]P(4 < X < 6) = P( -0.217 < Z< 0.6522[/tex]
=> [tex]P(4 < X < 6) = (Z< 0.6522) -P( Z < -0.217) [/tex]
From the z-table
[tex](Z< 0.6522) = 0.74286 [/tex]
So
[tex]P(4 < X < 6) = 0.74286 -0.4141[/tex]
=> [tex]P(4 < X < 6) = 0.74286 -0.4141[/tex]
=> [tex]P(4 < X < 6) = 0.32876[/tex]
Generally the probability that a randomly selected study participant's response was more than 8 is mathematically represented as
[tex]P(X > 8) = P(\frac{X - \mu }{\sigma} > \frac{8 - 4.5}{ 2.3} )[/tex]
[tex]P(X > 8) = P(Z > 1.52174 )[/tex]
From the z-table [tex]P(Z > 1.52174 ) = 0.064037[/tex]
So
[tex]P(X >8) = 0.064037 [/tex]
