Explanation:

Formula for the given probability is as follows.

           [tex]P(z > \frac{x – \mu}{\sigma})[/tex] = 0.10

       [tex]P(z > \frac{10.256 – \mu}{\sigma})[/tex] = 0.10

According to the normal table area we have,

           P(z < 1.28) = 0.10

Hence,  [tex]\frac{10.256 – \mu}{\sigma} = 1.28[/tex]

             [tex]\mu = 10.256 – 1.28 \times \sigma[/tex] ………. (1)

Also, the given probability is as follows.

           [tex]P(z < \frac{x – \mu}{\sigma})[/tex] = 0.05

          [tex]P(z < \frac{9.671 – \mu}{\sigma})[/tex] = 0.05

Hence,     [tex]\frac{9.671 – \mu}{\sigma} = -1.645[/tex]

               [tex]\mu = 9.671 + 1.645 \times \sigma[/tex] ……. (2)

Now, substitute the value of [tex]\mu[/tex] from equation (1) into equation (2) as follows.

      [tex]10.256 – 1.28 \times \sigma = 9.671 + 1.645 \times \sigma[/tex]    

      [tex]-2.925 \sigma = -0.585[/tex]

             [tex]\sigma[/tex] = 0.2

Putting the value of [tex]\sigma[/tex] into equation (2) we will find the value of [tex]\mu[/tex] as follows.

             [tex]\mu = 9.671 + 1.645 \times \sigma[/tex]

                       = [tex]9.671 + 1.645 \times 0.2[/tex]

                       = 10

Thus, we can conclude that the value of  [tex]\sigma[/tex] is 0.2 and the value of [tex]\mu[/tex] is 10.