Math Problem Statement

A manufacturer must test that his bolts are 4.00cm long when they come off the assembly line. He must recalibrate his machines if the bolts are too long or too short. After sampling 49 randomly selected bolts off the assembly line, he calculates the sample mean to be 3.87cm . He knows that the population standard deviation is 0.44cm . Assuming a level of significance of 0.02 , is there sufficient evidence to show that the manufacturer needs to recalibrate the machines? Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below.

H0Ha: μ=4.00: μ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯4.00

Solution

Step 1: State the null and alternative hypotheses.

The null hypothesis (H0H_0) and the alternative hypothesis (HaH_a) are stated as follows:

  • Null Hypothesis (H0H_0): The mean length of the bolts is equal to 4.00 cm.
    H0:μ=4.00H_0: \mu = 4.00

  • Alternative Hypothesis (HaH_a): The mean length of the bolts is not equal to 4.00 cm (indicating a two-tailed test).
    Ha:μ4.00H_a: \mu \neq 4.00

Would you like to proceed with calculating the test statistic and interpreting the results?

Here are 5 follow-up questions related to this topic:

  1. What does the level of significance (0.02) mean in this context?
  2. How is the test statistic calculated for this scenario?
  3. Why is a two-tailed test appropriate here instead of a one-tailed test?
  4. What are the critical values of the z-distribution for a significance level of 0.02?
  5. What assumptions are we making about the population and sampling process?

Tip: Always carefully determine whether a test should be one-tailed or two-tailed based on the research question or problem requirements.

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Math Problem Analysis

Mathematical Concepts

Hypothesis testing
Two-tailed test
Significance level
Population and sample means

Formulas

Null Hypothesis (H0): μ = μ0
Alternative Hypothesis (Ha): μ ≠ μ0
Z-test statistic: Z = (X̄ - μ) / (σ / √n)

Theorems

Central Limit Theorem
Properties of the Standard Normal Distribution

Suitable Grade Level

Grades 11-12 or College Introductory Statistics