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

Q: 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 $8.9 \mathrm{~kg}$.

∻Solution by Steps∻ ‖ step 1 ⋮ Understand the definitions of Type I and Type II errors in hypothesis testing. ‖ ‖ 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.◊What is the formula for calculating the probability of a Type II error in hypothesis testing?⍭ Generate me a similar question◊⚹

Question

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 $8.9 \mathrm{~kg}$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Understand the definitions of Type I and Type II errors in hypothesis testing. ‖ ‖ 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.◊What is the formula for calculating the probability of a Type II error in hypothesis testing?⍭ Generate me a similar question◊⚹

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