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

Step-by-step 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

       $P(X < 4) = P(\frac{X - \mu }{\sigma} < \frac{4 - 4.5}{ 2.3} )$

$\frac{X -\mu}{\sigma} = Z (The \ standardized \ value \ of\ X )$

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

     $P(4 < X < 6) = P( < \frac{4 - 4.5}{ 2.3} < \frac{X - \mu }{\sigma} < \frac{6 - 4.5}{2.3})$

=> $P(4 < X < 6) = P( \frac{4 - 4.5}{ 2.3} < \frac{X - \mu }{\sigma} < \frac{6 - 4.5}{2.3})$

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

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

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

       $P(X > 8) = P(\frac{X - \mu }{\sigma} > \frac{8 - 4.5}{ 2.3} )$

        $P(X > 8) = P(Z > 1.52174 )$

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

So

    $P(X >8) = 0.064037$