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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}
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