Question

According to a report from the United States Environmental Protection Agency, burning one gallon of gasoline typically emits about 8.9 kg of CO2. A fuel company wants to test a new type of gasoline designed to have lower CO2 emissions. Here are their hypotheses:

H0: μ = 8.9 kg
Ha: μ < 8.9 kg (where μ is the mean amount of CO2 emitted by burning one gallon of this new gasoline).

Which of the following would be a Type II error in this setting?

A. The mean amount of CO2 emitted by the new fuel is actually 89 kg but they conclude it is lower than 89 kg
B. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg
C. The mean amount of CO2 emitted by the new fuel is actual 89 kg and they alto conclude it is lower than 89 kg and they conclude it is lower than 9 kg
D. The mean amount of CO2 emitted by the new fuels actually lower than 8.9

Answer 1

Answer:

B. The mean amount of $CO_2$ emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg

Step-by-step explanation:

A Type II error is the failure to reject a false null hypothesis.

Given the null and alternate hypothesis of a fuel company which wants to test a new type of gasoline designed to have lower $CO_2$ emissions.:

$H_0: \mu = 8.9 kg\\H_a: \mu < 8.9 kg \\\text{ (where \mu is the mean amount of CO_2 emitted by burning one gallon of this new gasoline)}$

where $\mu$ is the mean amount of $CO_2$ emitted by burning one gallon of this new gasoline.

If the null hypothesis is false, then:

$H_a: \mu < 8.9 kg$

A rejection of the alternate hypothesis above will be a Type II error.

Therefore:

The Type II error is: (B) The mean amount of $CO_2$  emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

Answer 2

You can use the definition of type 2 error to find out which of the given option describes the type 2 error.

The Option which indicates Type II error is:

Option B. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

What is Type I and Type II error?

Firstly the whole story starts from hypotheses. The null hypothesis is tried to reject and we try to accept the alternate hypothesis.

• The type 1 error occurs if we get false positive conclusion (false positive means we accuse null hypothesis being wrong when it was actually correct).
• The type 2 error occurs if we get false negative conclusion (false negative means we accept null hypothesis when it was actually false).

The negative is just like the doctor's test getting negative means no disease. Similarly, if we conclude null hypothesis negative means it is accepted. If it is accepted wrongly means the negative test result was false, thus called false negative. This error is called type II error.

What is the type II error in the given context?

Since the null hypothesis here is $H_0: \text{typically emitted } {\rm CO_2} = 8.9 \: \rm kg$,

Thus the type II error would be when we fail to reject that CO2 emission is 8.9 kg (or say  we accept that CO2 emission is 8.9 kg generally) but the actual amount was lower than 8.9 kg(the alternate hypothesis was true)

Thus,

The Option which indicates Type II error is:

Option B. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

Learn more about type I and type II error here:

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