Answer:
(a) There is no evidence to support the claim that the two machines produce rods with different mean diameters.
P-value is 0.818
(b) 95% confidence interval for the difference in mean rod diameter is (-0.17, 0.27).
This interval shows that the difference in mean is between -0.17 and 0.27.
Step-by-step explanation:
(a) Null hypothesis: The two machines produce rods with the same mean diameter.
Alternate hypothesis: The two machines produce rods with different mean diameter.
Machine 1
mean = 8.73
variance = 0.35
n1 = 15
Machine 2
mean = 8.68
variance = 0.4
n2 = 17
pooled variance = [(15-1)0.35 + (17-1)0.4] ÷ (15+17-2) = 11.3 ÷ 30 = 0.38
Test statistic (t) = (8.73 - 8.68) ÷ sqrt[0.38(1/15 + 1/17)] = 0.05 ÷ 0.218 = 0.23
degree of freedom = n1+n2-2 = 15+17-2 = 30
Significance level = 0.05 = 5%
Critical values corresponding to 30 degrees of freedom and 5% significance level are -2.042 and 2.042.
Conclusion:
Fail to reject the null hypothesis because the test statistic 0.23 falls within the region bounded by the critical values -2.042 and 2.042.
There is no evidence to support the claim that the two machines produce rods with different mean diameters.
Cumulative area of test statistic is 0.5910
The test is a two-tailed test.
P-value = 2(1 - 0.5910) = 2×0.409 = 0.818
(b) Difference in mean = 8.73 - 8.68 = 0.05
pooled sd = sqrt(pooled variance) = sqrt(0.38) = 0.62
Critical value (t) = 2.042
E = t×pooled sd/√n1+n2 = 2.042×0.62/√15+17 = 0.22
Lower limit of difference in mean = 0.05 - 0.22 = -0.17
Upper limit of difference in mean = 0.05 + 0.22 = 0.27
95% confidence interval for the difference in mean rod diameter is between a lower limit of -0.17 and an upper limit of 0.27.
