Question

4) The ABC Company has recently replaced a machine critical to their manufacturing process. The average time for a worker to complete a certain task using the old machine was 16 mins. The times to complete the task using the machine are normally distributed with standard deviation 9 mins. The company wants to test if the average time to complete the task using the new machine has changed (i.e., is different). To this end, they timed 25 workers using the new machine to complete the task and determined their average time to be 13 mins. Answer the following questions to see what conclusion can be drawn from the sample evidence at a 1% significance level. :5 a) State the null and the alternative hypotheses b) What type of test is this (left-tailed, right-tailed, two-tailed)? Why? c) Write the formula for the test-statistic to be used for this test. d) What is the distribution of the test-statistic? e) Draw the density curve of the distribution and indicate the rejection and non-rejection regions How much area is under the curve in the rejection region(s)? f Write the rejection region(s) in interval notation. g) Compute the value of the test-statistic using the relevant information given in the problem h) Compute the p-value for the test. i) Do we reject or fail to reject the null hypothesis at the specified (1%) significance level? j) Write your overall conclusion in one complete sentence.

Answer 1

a) The hypotheses are

Ho : μ = 16 H1 :11 16

b) This is a two tailed z-test, since we are testing whether the mean is different from 16.

Since the population standard deviation is known we use z-distribution.

c) The test statistic is

The test statistic has z-distribution.

e) The critical value is

0.995

f) The rejection region is

z <ー2.575829 > 2.575829

g) The test statistic is

X-55 z= 13 16 9 V25 Z-5

h) The P-value is

p-value-20(-5) 0.00000057

i) We reject the null hypothesis since

j) We conclude that the mean time to complete the task with the new machine is different from 16.