Question

Math

Posted 4 months ago

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According to a report from the United States Environmental Protection Agency, burning one gallon of gasoline typically emits about $8.9 \mathrm{~kg}$ of $\mathrm{CO}_{2}$. A fuel company wants to test a new type of gasoline designed to have lower $\mathrm{CO}_{2}$ emissions. Here are their hypotheses:
$\begin{array}{c}
H_{0}: \mu=8.9 \mathrm{~kg} \\
H_{\mathrm{a}}: \mu<8.9 \mathrm{~kg}
\end{array}$
(where $\mu$ is the mean amount of $\mathrm{CO}_{2}$ emitted by burning one gallon of this new gasoline).
Which of the following would be a Type II error in this setting?
Choose 1 answer:
(A) The mean amount of $\mathrm{CO}_{2}$ emitted by the new fuel is actually $8.9 \mathrm{~kg}$, and they conclude it is lower than $8.9 \mathrm{~kg}$.
(B) The mean amount of $\mathrm{CO}_{2}$ emitted by the new fuel is actually lower than $8.9 \mathrm{~kg}$, and they fail to conclude it is lower than $8.9 \mathrm{~kg}$.
(C) The mean amount of $\mathrm{CO}_{2}$ emitted by the new fuel is actually $8.9 \mathrm{~kg}$, and they fail to conclude it is lower than $8.9 \mathrm{~kg}$.
(D) The mean amount of $\mathrm{CO}_{2}$ emitted by the new fuel is actually lower than $8.9 \mathrm{~kg}$, and they conclude it is lower than $8.9 \mathrm{~kg}$.
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

A Type I error occurs when the null hypothesis is true, but we incorrectly reject it

step 3

A Type II error occurs when the null hypothesis is false, but we fail to reject it

step 4

In the context of the given hypotheses, a Type II error would happen if the true mean CO₂ emission is less than 8.9 kg (making the null hypothesis false), but the test fails to reject the null hypothesis

step 5

Option (B) describes a scenario where the mean CO₂ emission is actually lower than 8.9 kg, but the test fails to conclude this, which is a Type II error

B

Key Concept

Type II Error

Explanation

A Type II error in hypothesis testing occurs when the null hypothesis is false, but we fail to reject it.

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