Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 6.46 millimeters and a standard deviation of 0.05 millimeters. Find the two diameters that separate the top 9% and the bottom 9%. 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 9% and the bottom 9% are 6.53 and   6.39 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 6.46 millimeters and a standard deviation of 0.05 millimeters.

Let X = diameters of bolts produced in a machine shop

So, X ~ N($\mu=6.46,\sigma^{2} = 0.05^{2}$)

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 9% and the bottom 9%.

• Firstly, Probability that the diameter separate the top 9% is given by;

          P(X > x) = 0.09

          P( $\frac{X-\mu}{\sigma}$ > $\frac{x-6.46}{0.05}$ ) = 0.09

          P(Z > $\frac{x-6.46}{0.05}$ ) = 0.09

So, the critical value of x in z table which separate the top 9% is given as 1.3543, which means;

                     $\frac{x-6.46}{0.05}$ = 1.3543

                     $x-6.46 = 0.05 \times 1.3543$

                        $x$ = 6.46 + 0.067715 = 6.53

• Secondly, Probability that the diameter separate the bottom 9% is given by;

          P(X < x) = 0.09

          P( $\frac{X-\mu}{\sigma}$ < $\frac{x-6.46}{0.05}$ ) = 0.09

          P(Z < $\frac{x-6.46}{0.05}$ ) = 0.09

So, the critical value of x in z table which separate the bottom 9% is given as -1.3543, which means;

                     $\frac{x-6.46}{0.05}$ = -1.3543

                     $x-6.46 = 0.05 \times (-1.3543)$

                        $x$ = 6.46 - 0.067715 = 6.39

Therefore, the two diameters that separate the top 9% and the bottom 9% are 6.53 and 6.39 respectively.

Answer 2

Answer:

Step-by-step explanation:

Let X be the diameter of bolts produced in a machine shop

X is normal with mean = 6.46 mm and std dev = 0.05 mm.

We are to find the two diameters that separate the top 9% and the bottom 9%.

We can use std normal distribution value for this and from Z we can calculate X values

P(Z<z) = 0.09 and P(Z<z1) = 0.91

z=-1.341 and z1 = +1.341

Corresponding X values would be

for bottom 9%

X = 6.46-1.341*0.05 = 6.39295

X = 6.46+1.341*0.05=6.52305