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QUESTION The diameters of ball bearings produced by a process are
normally distributed with standard deviation fo 0.04mm. A random
sample of 7 is taken and their measurements are:

7.99\ 8.01\ 7.98\ 8.07\ 8.00\ 8.02\ 7.93\ mm.

Calculate a $95\%$ confidence interval for $\mu$, the true mean
diameter of the ball bearings produced by this process.

The process is modified in a way that is likely to change both the
mean and the standard deviation of the ball bearings produced.  A
random sample of 6 is taken and the measurements are

8.03\ 8.09\ 7.94\ 7.89\ 8.15\ 8.08

Calculate a $95\%$ confidence interval for $\mu$, the new mean
diameter.  Test whether the standard deviation has been changed by
the new process.


ANSWER

  \begin{tabular}{ccccccc}
  7.99&8.01&7.98&8.07&8.00&8.02&7.93
  \end{tabular}

  $\overline{x}=8.00\ \ n=7\ \ \sigma=0.04\\
  95 \% CI\ \ 8.03\pm 2.571 \times \frac{0.0982}{\sqrt{6}}=8.03
  \pm 0.103\\
  H_0:\sigma^2=(0.04)^2\ \ H_1: \sigma^2 \neq (0.04)^2\ \ \alpha =
  5\%$

  Test of single variance is test 3. Assume normal distribution,
  z$=\frac{(n-1)s^2}{\sigma _0^2}\sim \chi_{n-1}^2\\
  z=\frac{5 \times 0.098262}{0.0462}=30.125\ \ s=0.0982\ \ n=6$

  $\begin{array}{c}
   \textrm{Hence reject }H_0 \textrm{ and}\\
   \textrm{accept standard}\\
   \textrm{deviation changed}.
   \end{array}
   \ \ \
   \begin{array}{c}
   \epsfig{file=641-5-2.eps, width=70mm}
   \end{array}$


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