Let��s assess whether the null and alternative hypotheses (H₀ and H��) are correctly set up.

General guideline for null and alternative hypotheses:

• Null hypothesis (H₀): A statement of no effect, no difference, or a specific claim (usually includes "=").
• Alternative hypothesis (H₁): A statement that contradicts H₀, suggesting that there is an effect, difference, or that the value is not equal to the null claim (can include "≠", ">", "<", etc.).

Now, let's analyze the pairs:

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1. H₀: M = 30
   H₁: M ≠ 30

• Analysis:
  • The null hypothesis states that the parameter $M$ (typically a mean or median) is equal to 30.
  • The alternative hypothesis contradicts H₀ by stating that $M$ is not equal to 30.
  • Verdict: Valid — This follows the correct structure for a two-tailed test.

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2. H₀: μ = 115
   H₁: μ ≠ 115

• Analysis:
  • The null hypothesis states that the population mean $μ$ is 115.
  • The alternative hypothesis states that $μ$ is not equal to 115.
  • Verdict: Valid — This is the correct setup for a two-tailed test.

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3. H₀: μ = 210
   H₁: μ ≠ 210

• Analysis:
  • The null hypothesis assumes that the population mean $μ$ is equal to 210.
  • The alternative hypothesis suggests that the mean is not equal to 210.
  • Verdict: Valid — Again, this is correctly set up for a two-tailed test.

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4. H₀: µ = 130
   H₁: µ ≠ 138

• Analysis:
  • The null hypothesis states that the population mean $μ$ is 130.
  • The alternative hypothesis states that $μ$ is not equal to 138.
  • This is incorrect because the alternative hypothesis must contradict the null hypothesis by involving the same value.
  • Verdict: Invalid — The correct alternative hypothesis should be $μ \neq 130$, not $μ \neq 138$.

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Thus, the results are:

1. Valid
2. Valid
3. Valid
4. Invalid

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Would you like to explore further details, or do you have any questions? Here are some related questions for further study:

1. What is the difference between a one-tailed and two-tailed hypothesis test?
2. How is the significance level used in hypothesis testing?
3. Why do we always assume the null hypothesis is true when conducting a hypothesis test?
4. Can the null hypothesis contain inequalities like ">" or "<"?
5. How do you determine whether to use a one-sample or two-sample hypothesis test?

Tip: In hypothesis testing, always ensure that the null and alternative hypotheses involve the same population parameter (e.g., mean, proportion) with the null typically stating equality and the alternative contradicting it.