Question

In a recent study on world​ happiness, participants were asked to evaluate their current lives on a scale from 0 to​ 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally​ distributed, with a mean of

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

`P(X < 4) =  0.4141`

b

`P(4 < X < 6) =   0.32876`

c

`P(X >8) =  0.064037`

d

The correct option is B

Explanation:

From the question we are told that

The mean is
`\mu =  4.5`

The standard deviation is
`\sigma  =  2.3`

Generally the probability that a randomly selected study participant's response was less than 4 is mathematically represented as

`P(X < 4) =  P((X - \mu )/(\sigma) < (4 - 4.5)/( 2.3)   )`

`(X -\mu)/(\sigma) =  Z (The \  standardized \  value \  of\  X )`

So

`P(X < 4) =  P(Z < -0.217  )`

From the z-table
`P(Z < -0.217  ) =  0.4141`

So

`P(X < 4) =  0.4141`

Generally the probability that a randomly selected study participant's response was between 4 and 6 is mathematically represented as

`P(4 < X < 6) =  P( < (4 - 4.5)/( 2.3)  < (X - \mu )/(\sigma) < (6 - 4.5)/(2.3))`

=>
`P(4 < X < 6) =  P(  (4 - 4.5)/( 2.3)  < (X - \mu )/(\sigma) < (6 - 4.5)/(2.3))`

=>
`P(4 < X < 6) =  P(  -0.217  < Z< 0.6522`

=>
`P(4 < X < 6) =   (Z< 0.6522) -P( Z <  -0.217)`

From the z-table

`(Z< 0.6522)  = 0.74286`

So

`P(4 < X < 6) =  0.74286 -0.4141`

=>
`P(4 < X < 6) =  0.74286 -0.4141`

=>
`P(4 < X < 6) =   0.32876`

Generally the probability that a randomly selected study participant's response was more than 8 is mathematically represented as

`P(X > 8) =  P((X - \mu )/(\sigma) > (8 - 4.5)/( 2.3)   )`

`P(X > 8) =  P(Z > 1.52174  )`

From the z-table
`P(Z > 1.52174 ) =  0.064037`

So

`P(X >8) =  0.064037`