Question

Clint Barton is a self-employed consultant who runs his company, Hawkeye Solutions, out of a house that he owns that he does not live in. He uses the main floor for his work and rents out the second floor to a tenant. Since he uses a fair amount of computing in his work, he wants to make sure that he writes off a representative portion of the electrical bill for the house against the business. In the past, he estimated that 70% of the electricity in the house goes towards the business.

Since the Canada Revenue Agency might want to see documentation about his expenses, Barton wants to sample some of the power use to provide support for his case. Since the building has central heating and cooling, these electrical costs are shared evenly as utilities and Barton has already accounted for them separately.

Barton connected monitors to lines going to each floor for non-utility power. Because he has had disagreements about proper expenses in the past, he wants to provide strong evidence to the Canada Revenue Agency to support his claim that he uses 70% of the non-utility power for the business.

Design a test for Barton where he will record the values of the power going upstairs and to the main floor every day for a month (30 days). And answer the following questions (use complete sentences and exact equations where possible):

Barton does the test as you outline above. He finds over the course of the sample that the business used $320 of power and the upstairs used $80.

f) What is the value of the test statistic in this case?

g) What is the decision of the test in this case? Explain your reasoning.

h) What does this mean for Barton’s attempt to get sufficient documentation?

A month later, Barton does a test with a sample of 2 months (60 days). He finds over the course of the sample that the business used $720 of power and the upstairs used $180.

i) What is the value of the test statistic in this case?

j) What is the decision of the test in this case? Explain your reasoning.  

k) What does this mean for Barton’s attempt to get sufficient documentation?  

Answer 1

Solution

Let p represent the actual proportion of the non-utility power used for the business.

The question revolves around testing whether this p is 0.7 (i.e., 70%) or not.

Or, put in statistical terms, we want to test:

Hypotheses:

Null H₀ : p = p₀   = 0.7 Vs Alternative H_A : p ≠ 0.7

Part (f)

Test Statistic:

Z = (p_hat - p₀)/√{p₀(1 - p₀)/n} = 1.1952 Answer 1

Where

p_hat = sample proportion = (320)/( 320 + 80) = 0.8

and

n = sample size = 30.

Calculations:

n	30
phat	0.8
Zcal	1.1952
α(assumed)	0.05
Zcrit	1.9600
p-value	0.2320

Part (g)

Decision:

The null hypothesis is accepted. Answer 2

Back-up Reasoning:

Distribution, Significance Level, α Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper (α/2)% of N(0, 1), and

p-value = P(Z > | Zcal |)

Using Excel Function: Statistical NORMINV and NORMDIST these are found as shown in the above table.

Decision:

Since | Zcal |< Zcrit, or equivalently, since p-value > α, H₀ is accepted.

Conclusion :

There is enough evidence to suggest that the claim that the actual proportion of the non-utility power used for the business. Is 70% holds valid.

Part (h)

Based on the above answer, Barton can establish that the proportion of the non-utility power used for the business is indeed 70%. Answer 3

Part (i)

Test Statistic:

Z = (p_hat - p₀)/√{p₀(1 - p₀)/n} = 1.6903 Answer 4

Where

p_hat = sample proportion = (720)/( 720 + 180) = 0.8

and

n = sample size = 30.

Calculations:

n	60
phat	0.8
Zcal	1.6903
α(assumed)	0.05
Zcrit	1.9600
p-value	0.2320

Part (j)

Decision:

The null hypothesis is accepted. Answer 5

Back-up Reasoning:

Distribution, Significance Level, α Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper (α/2)% of N(0, 1), and

p-value = P(Z > | Zcal |)

Using Excel Function: Statistical NORMINV and NORMDIST these are found as shown in the above table.

Decision:

Since | Zcal |< Zcrit, or equivalently, since p-value > α, H₀ is accepted.

Conclusion :

There is enough evidence to suggest that the claim that the actual proportion of the non-utility power used for the business. Is 70% holds valid.

Part (k)

Based on the above answer, Barton can reaffirm that the proportion of the non-utility power used for the business is indeed 70%. Answer 6

DONE