Solved on Jan 05, 2024

Redefine a claim's complement and identify $H_0$ and $H_a$ . The claim is $\mu \geq 508$ . The complement is $\mu < 508$ . $H_0: \mu \geq 508$ , $H_a: \mu < 508$ .

STEP 1

Assumptions
1. We are given a claim about a population mean $\mu$ .
2. The claim is that $\mu \geq 508$ .
3. We need to write the complement of this claim.
4. We need to identify which statement is the null hypothesis $\mathrm{H}_{0}$ and which is the alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ .

STEP 2

The null hypothesis $\mathrm{H}_{0}$ is a statement of no effect or no difference, and it is the hypothesis that is initially assumed to be true. The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is what we are trying to find evidence for in our hypothesis test.

STEP 3

The complement of a claim is the statement that represents the opposite of the original claim. If the original claim is $\mu \geq 508$ , the complement will be $\mu < 508$ .

STEP 4

Write the complement of the claim.
$\mu < 508$

STEP 5

Now, we need to decide which is $\mathrm{H}_{0}$ and which is $\mathrm{H}_{\mathrm{a}}$ . The null hypothesis is typically a statement of equality or no effect, which in this case is represented by the equality part of the original claim $\mu \geq 508$ .

STEP 6

Identify the null hypothesis $\mathrm{H}_{0}$ .
$\mathrm{H}_{0}: \mu = 508$

STEP 7

The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is the complement of the null hypothesis and represents the inequality part of the original claim.

STEP 8

Identify the alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ .
$\mathrm{H}_{\mathrm{a}}: \mu < 508$

STEP 9

Determine which option correctly lists $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$ .
Option E states $\mathrm{H}_{0}: \mu \leq 508$ , which is not the correct form for the null hypothesis since it includes an inequality. The null hypothesis should be a statement of equality.
Option B states $\mathrm{H}_{0}: \mu = 508$ , which is the correct form for the null hypothesis.
Option C states $\mathrm{H}_{\mathrm{a}}: \mu \neq 508$ , which is not the correct alternative hypothesis for our claim since it suggests a two-sided alternative, whereas our claim leads to a one-sided alternative.
The correct option that lists both $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$ is:
B. $\mathrm{H}_{0}: \mu = 508$ C. $\mathrm{H}_{\mathrm{a}}: \mu < 508$

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Redefine a claim's complement and identify $H_0$ and $H_a$ . The claim is $\mu \geq 508$ . The complement is $\mu < 508$ . $H_0: \mu \geq 508$ , $H_a: \mu < 508$ .

STEP 1

STEP 2

The null hypothesis $\mathrm{H}_{0}$ is a statement of no effect or no difference, and it is the hypothesis that is initially assumed to be true. The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is what we are trying to find evidence for in our hypothesis test.

STEP 3

The complement of a claim is the statement that represents the opposite of the original claim. If the original claim is $\mu \geq 508$ , the complement will be $\mu < 508$ .

STEP 4

Write the complement of the claim.
$\mu < 508$

STEP 5

Now, we need to decide which is $\mathrm{H}_{0}$ and which is $\mathrm{H}_{\mathrm{a}}$ . The null hypothesis is typically a statement of equality or no effect, which in this case is represented by the equality part of the original claim $\mu \geq 508$ .

STEP 6

Identify the null hypothesis $\mathrm{H}_{0}$ .
$\mathrm{H}_{0}: \mu = 508$

STEP 7

The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is the complement of the null hypothesis and represents the inequality part of the original claim.

STEP 8

Identify the alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ .
$\mathrm{H}_{\mathrm{a}}: \mu < 508$

STEP 9

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Redefine a claim's complement and identify H0H_0H0​ and HaH_aHa​. The claim is μ≥508\mu \geq 508μ≥508. The complement is μ<508\mu < 508μ<508. H0:μ≥508H_0: \mu \geq 508H0​:μ≥508, Ha:μ<508H_a: \mu < 508Ha​:μ<508.

STEP 1

STEP 2

The null hypothesis H0\mathrm{H}_{0}H0​ is a statement of no effect or no difference, and it is the hypothesis that is initially assumed to be true. The alternative hypothesis Ha\mathrm{H}_{\mathrm{a}}Ha​ is what we are trying to find evidence for in our hypothesis test.

STEP 3

The complement of a claim is the statement that represents the opposite of the original claim. If the original claim is μ≥508\mu \geq 508μ≥508, the complement will be μ<508\mu < 508μ<508.

STEP 4

Write the complement of the claim.μ<508\mu < 508μ<508

STEP 5

Now, we need to decide which is H0\mathrm{H}_{0}H0​ and which is Ha\mathrm{H}_{\mathrm{a}}Ha​. The null hypothesis is typically a statement of equality or no effect, which in this case is represented by the equality part of the original claim μ≥508\mu \geq 508μ≥508.

STEP 6

Identify the null hypothesis H0\mathrm{H}_{0}H0​.H0:μ=508\mathrm{H}_{0}: \mu = 508H0​:μ=508

STEP 7

The alternative hypothesis Ha\mathrm{H}_{\mathrm{a}}Ha​ is the complement of the null hypothesis and represents the inequality part of the original claim.

STEP 8

Identify the alternative hypothesis Ha\mathrm{H}_{\mathrm{a}}Ha​.Ha:μ<508\mathrm{H}_{\mathrm{a}}: \mu < 508Ha​:μ<508

STEP 9

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Redefine a claim's complement and identify $H_0$ and $H_a$ . The claim is $\mu \geq 508$ . The complement is $\mu < 508$ . $H_0: \mu \geq 508$ , $H_a: \mu < 508$ .

The null hypothesis $\mathrm{H}_{0}$ is a statement of no effect or no difference, and it is the hypothesis that is initially assumed to be true. The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is what we are trying to find evidence for in our hypothesis test.

The complement of a claim is the statement that represents the opposite of the original claim. If the original claim is $\mu \geq 508$ , the complement will be $\mu < 508$ .

Write the complement of the claim.
$\mu < 508$

Now, we need to decide which is $\mathrm{H}_{0}$ and which is $\mathrm{H}_{\mathrm{a}}$ . The null hypothesis is typically a statement of equality or no effect, which in this case is represented by the equality part of the original claim $\mu \geq 508$ .

Identify the null hypothesis $\mathrm{H}_{0}$ .
$\mathrm{H}_{0}: \mu = 508$

The alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ is the complement of the null hypothesis and represents the inequality part of the original claim.

Identify the alternative hypothesis $\mathrm{H}_{\mathrm{a}}$ .
$\mathrm{H}_{\mathrm{a}}: \mu < 508$