Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 4% and the bottom 4%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

Answer:

Two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.

Step-by-step explanation:

We are given that the diameters of bolts produced in a machine shop are normally distributed with a mean of 5.65 millimeters and a standard deviation of 0.07 millimeters.

Let X = diameters of bolts produced in a machine shop

So, X ~ N($\mu=5.65,\sigma^{2} = 0.07^{2}$)

The z score probability distribution is given by;

         Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = population mean

            $\sigma$ = standard deviation

Now, we have to find the two diameters that separate the top 4% and the bottom 4%.

• Firstly, Probability that the diameter separate the top 4% is given by;

        P(X > x) = 0.04

        P( $\frac{X-\mu}{\sigma}$ > $\frac{x-5.65}{0.07}$ ) = 0.04

        P(Z > $\frac{x-5.65}{0.07}$ ) = 0.04

So, the critical value of x in z table which separate the top 4% is given as 1.7507, which means;

                 $\frac{x-5.65}{0.07}$  = 1.7507

              $x-5.65 = 0.07 \times 1.7507$

                         $x$ = 5.65 + 0.122549 = 5.77

• Secondly, Probability that the diameter separate the bottom 4% is given by;

        P(X < x) = 0.04

        P( $\frac{X-\mu}{\sigma}$ < $\frac{x-5.65}{0.07}$ ) = 0.04

        P(Z < $\frac{x-5.65}{0.07}$ ) = 0.04

So, the critical value of x in z table which separate the bottom 4% is given as -1.7507, which means;

                 $\frac{x-5.65}{0.07}$  = -1.7507

              $x-5.65 = 0.07 \times (-1.7507)$

                         $x$ = 5.65 - 0.122549 = 5.53

Therefore, the two diameters that separate the top 4% and the bottom 4% are 5.77 and 5.53 respectively.