Question

The Umbrella Corporation has companies in many sectors of our economy. One of their paint manufacturing plants sells cans of paint and uses a machine that has been calibrated to fill the cans to contain an average of 128 ounces each. To test whether their machine needs calibration, the manufacturer takes a random sample of 37 cans and finds they average 129.5 ounces with a standard deviation of 3.55 ounces. Is there sufficient evidence that the filling machine is set too high (filling the cans with too much paint)? (Assume all assumptions are met)

Answer 1

Test: μ > 128 with normal dist" Hypothesis test Ho : μ 128 Ha : μ > 128 (Null Hypothesis, Ho : μ = 128 is also correct) (Alternative Hypothesis, also called H1) This is upper tailed test (one tailed test) Since sample size is large, sample standard deviation can be taken as an appror- imation of population standard deviation 129.5 σ = 3.55 n-37 Significance Level, α = 0.05 Test statistic z* 129.5 - 128 Test Statistic、s*- 2.570 181 ~ 2.57 3551 Test Statistic, * 2.57 P-value is in direction of Alternative hypothesis Since Ha : μ > 128, P-value-P(Z > 2.57) P-value is Area to the right of test statistic = 2.57 P-value = 0.0051 test statistic 2.57

P-value = P(Z > 2.57) = 0.0051 (from z-table) P-value = 0.0051 Decision Rule (rejection criteria): Reject Ho if p-value < a Decision: Since 0.0051<0.05, we reject the null hypothesis. Using excel function normdist -(129.5- 128)/( 3.55/sqrt(37))),0,1, TRUE) or TI's function normalcdf( 1E-99, -( (129.5- 128)/(3.55/sqrt(37)) answer is: 0.0050822645 The Critical Value-Zorit-«oー2005-1645 (From z table , using interpola- tion, 1/2 th distance between 1.64 and 1.65 Critical value 1.645 Decision Rule (rejection criteria): Reject Ho if 2 tailed test) (because this is upper Rejection region has been shaded: 0.05 a 1.645 Decision: Since 2.57> 1.645, we (reject the null hypothesis. Hence at 5% significance level, we have sufficient evidence to conclude that the filling machine is set too high (filling the cans with too much paint) ~e Thank you