Question

During the manufacturing of commercial windows, hot steel ingots are passed though a rolling process and flattened to a prescribed thickness. When the manufacturing process is working properly the ingots should have a mean thickness of 0.75 inches. A random sample of 40 ingots is taken and the mean was found to be 0.725 inches with a standard deviation of 0.092.

a. Find a 95% confidence interval for the population mean thickness.

b. Find the error bound.

c. Interpret the confidence interval in the words of the problem.

d. Is there evidence that the machine is not functioning properly? Justify!

Answer 1

a) Sample Mean-T = 0.725 Since sample size is large, sample standard deviation can be taken as an approx- imation of population standard deviation. Population Standard deviation = σ = 0.092 Sample Size = n-40 Significance level-a-1-Confidence-1-0.95-0.05 The Critical Value-_a/2 = z0.025- 1.96 (From z table) Critical Values ?n/2 =士1.96 Margin of Error-E-4/2 × -1.96 × 0.092 1 0.02851 Limits of 95% confidence interval are given by: Lower limit E-0.725-0.02851 0.696490 Upper limit-T + E = 0.725 + 0.02851 0.753510 0.696 0.754 95% confidence interval is: x E-0.725士0.02851 (0.69649, 0.75351) 95% CI using normal dist: 0.696 < μ < (0.754 Also, the exact answer using TI-83/84 or excel is 0.725± invNorm(0.975) × 0.092/sqrt(40) -(0.6964903639, 0.7535096361) b Margin of Error, E 0.0285 Interpretation: We are 95% sure that the interval contains the true population mean. d) Since the CI includes 0.75 within the limits we conclude that the mahchin<e is function properly @seoThank บู01LJO