Question

A manufacturer produces pistons rings for an automobile engine. It is known that the ring is approximately normally distributed and has standard deviation of s = 0.001 mm. A random sample of 15 rings has a mean diameter of ?=74.036 x ̅=74.036 mm.

(a) Using a level of significance of 0.01, state and test the hypothesis that the mean piston ring diameter is 74.035 mm.

(b) What is the minimum level of significance to reject the null hypothesis?

c) Use the appropiate confidence interval to test the hypothesis and draw conclusions

Answer 1

a) H₀: = 74.035

    H₁: 74.035

The test statistic t = ()/(s/)

                            = (74.036 - 74.035)/(0.001/)

                            = 3.873

At alpha = 0.01, the critical values are t^* = +/- 2.977

Since the test statistic value is greater than the positive critical value(3.873 > 2.977), so we should reject the null hypothesis.

So there is not sufficient evidence to conclude that the mean piston ring diameter is 74.035 mm.

b) P-value = 2 * P(T > 3.873)

                 = 2 * (1 - P(T < 3.873))

                 = 2 * (1 - 0.9992) = 0.0016

So the minimum level of significance to reject the null hypothesis is 0.0016.

c) The 99% confidence interval is

+/- t^* * s/

= 74.036 +/- 2.977 * 0.001/

= 74.036 +/- 0.0008

= 74.0352, 74.0368

Since the interval does not contain the hypothesized value 74.035, so we should reject the null hypothesis.