Question

(The life in hours of a heating element used in a furnace is known to be approximately normally distributed. A random sample of 15 heating elements is selected and found to have an average life of 598.14 hours and a sample standard deviation of 16.93 hours.

(a) Test the claim that the true average life is greater than 550 hours. Be sure to state Hypotheses clearly, give all summary data, state Test used, show Test Statistics formula and value, and show how to find the P-value. Write an appropriate conclusion using α = 0.05. 7

(b) Test the claim that the true average life is equal to 610 hours. Be sure to state Hypotheses clearly, give all summary data, state Test used, show Test Statistics formula and value, and show how to find the P-value. Write an appropriate conclusion using α = 0.01.

Answer 1

Answer)

As the population standard deviation is unknown we need to use standard normal z table to conduct the test

A)

Null hypothesis Ho : u = 550

Alternate hypothesis Ha : u > 550

Test statistics z = (sample mean - claimed mean)/(s.d/√n)

Z = (598.14 - 550)/(16.93/√15) = 11.013

Degrees of freedom is = n-1 = 14

For 14 dof and 11.013 test statistics, p-value from t distribution is = 0

As the obtained p-value is less than the given significance 0.05

We reject the null hypothesis

So, we have enough evidence to conclude thar u > 550

B)

Null hypothesis Ho : u = 610

Alternate hypothesis Ha : u not equal to 610

Test statistics t = (598.14-610)/(16.93/√15) = -2.713

Degrees of freedom is = 14

For 14 dof and -2.713 test statistics

P-value = 0.0168

As the obtained p-value is greater than 0.01 (given significance)

We fail to reject the null hypothesis

So, we have enough evidence to conclude that u is equal to 610