Question 1178776: A study claims that all homeowners in a town spend an average of 8 hours or
more on house cleaning and gardening during a weekend. A researcher
wanted to check if this claim is true. A random sample of 20 homeowners
taken by this researcher showed that they spend an average of 7.68 hours on
such chores during a weekend. The population of such times for all
homeowners in this town is normally distributed with the population standard
deviation of 2.1 hours. Using the 1% significance level, can you conclude that
the claim that all homeowners spend an average of 8 hours or more on such
chores during a weekend is false? [10 marks]
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's conduct a hypothesis test to determine if the claim is false.
**1. Define the Hypotheses:**
* **Null Hypothesis (H₀):** μ ≥ 8 (The average time spent is 8 hours or more)
* **Alternative Hypothesis (H₁):** μ < 8 (The average time spent is less than 8 hours)
This is a left-tailed test.
**2. Set the Significance Level:**
* α = 0.01 (1%)
**3. Given Data:**
* Sample size (n): 20
* Sample mean (x̄): 7.68 hours
* Population standard deviation (σ): 2.1 hours
**4. Calculate the Test Statistic (z-score):**
* z = (x̄ - μ) / (σ / √n)
* z = (7.68 - 8) / (2.1 / √20)
* z = -0.32 / (2.1 / 4.472)
* z = -0.32 / 0.4696
* z ≈ -0.6814
**5. Find the Critical Value:**
* For a left-tailed test with α = 0.01, the critical z-value is -2.33 (from a z-table or calculator).
**6. Make a Decision:**
* Compare the calculated z-score (-0.6814) to the critical z-value (-2.33).
* Since -0.6814 > -2.33, the calculated z-score does not fall in the rejection region.
* Therefore, we fail to reject the null hypothesis.
**7. Draw a Conclusion:**
* There is not sufficient evidence at the 1% significance level to conclude that the claim that all homeowners spend an average of 8 hours or more on such chores during a weekend is false.
**In summary:**
The research does not provide enough evidence to reject the claim, so we cannot conclude that the claim is false.
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