Math Problem Statement

H0: m=4 versus H1: m≠4. Since the 95% confidence interval (Blank: A. contains, B. does not contain) m0, then H0 (Blank: A. will, B. will not) be rejected at the 0.05 level.

Solution

To answer this question:

Key Points to Consider:

  1. Hypotheses:

    • Null Hypothesis (H0H_0): m=4m = 4
    • Alternative Hypothesis (H1H_1): m4m \neq 4
  2. Confidence Interval Relationship:

    • The 95% confidence interval reflects a range of plausible values for the population mean (mm) based on sample data.
    • If the hypothesized value (m0=4m_0 = 4) lies inside the confidence interval, we fail to reject H0H_0.
    • If the hypothesized value (m0=4m_0 = 4) lies outside the confidence interval, we reject H0H_0.
  3. Significance Level:

    • At a 0.050.05 level, a 95% confidence interval corresponds directly to the hypothesis test.

Completing the Sentence:

  1. If the confidence interval contains m0=4m_0 = 4:

    • Select A. contains.
    • Then, H0H_0 will not be rejected at the 0.050.05 level (select B. will not).
  2. If the confidence interval does not contain m0=4m_0 = 4:

    • Select B. does not contain.
    • Then, H0H_0 will be rejected at the 0.050.05 level (select A. will).

Final Selection Based on Confidence Interval:

  • If you determine the confidence interval relative to m0m_0, substitute the appropriate answers into the blanks.

Would you like additional clarification or details? Let me know! Here are related questions to expand your understanding:

  1. What is the relationship between confidence intervals and p-values in hypothesis testing?
  2. How do we calculate a confidence interval for a given dataset?
  3. What does it mean to set a two-tailed hypothesis test, as in this example?
  4. How does the confidence level (e.g., 95%) influence the hypothesis test decision?
  5. Can a result be significant at a lower confidence level but not at 95%?

Tip: Always visualize the confidence interval on a number line to check if the hypothesized value falls within or outside the range.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Confidence Intervals
Significance Levels

Formulas

Confidence interval formula
Relationship between confidence intervals and hypothesis testing

Theorems

Central Limit Theorem
Concept of p-value in hypothesis testing

Suitable Grade Level

Undergraduate (Statistics or Data Analysis Courses)