Question

The owners of an industrial plant want to determine which of two types of fuel (gas or electricity) will produce more useful energy at a lower cost. The cost is measured by plant investment per delivered quad ($ invested /quadrillion BTUs). The smaller this number, the less the industrial plant pays for delivered energy. Suppose we wish to determine if there is a difference in the average investment/quad between using electricity and using gas. Our null and alternative hypotheses would be:

a. H0: (?e - ?g) = 0 vs. Ha: (?e - ?g) = 0
b. H0: (?e - ?g) = 0 vs. Ha: (?e - ?g) <0
c. H0: (?e - ?g) = 0 vs. Ha: (?e - ?g) > 0
d. H0: (?e - ?g) = 0 vs. Ha: (?e - ?g) ?0

Answer 1

d. H0:
`(\mu_e - \mu_g) = 0` vs.
`Ha: (\mu_e - \mu_g) \\eq 0`

Explanation:

A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".

The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".

The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".

On this case the Management try to determine if there is a difference in the average investment/quad between using electricity and using gas. So this needs to be on the alternative hypothesis. And the complement of the alternative hypothesis would be on the null hypothesis.

Null hypothesis :
`\mu_e = \mu_g`

Alternative hypothesis:
`\mu_e \\eq \mu_g`

Or equivalently:

Null hypothesis :
`\mu_e - \mu_g = 0`

Alternative hypothesis:
`\mu_e - \mu_g \\eq 0`

Other important thing is that on the alternative hypothesis we never can have the symbol equal.

So on this case the best options is:

d. H0:
`(\mu_e - \mu_g) = 0` vs.
`Ha: (\mu_e - \mu_g) \\eq 0`