Question 1168856: a sample of 400 male students is found to have a mean height of 67.47 inches.can it be reasonably regarded as a sample from a large population with mean height 67.39 inches and standard deviation 1.3 inches?test at 5% level of significance
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this hypothesis test step-by-step.
**1. Define the Hypotheses**
* **Null Hypothesis (H₀):** The sample mean height is equal to the population mean height.
* H₀: µ = 67.39 inches
* **Alternative Hypothesis (H₁):** The sample mean height is not equal to the population mean height.
* H₁: µ ≠ 67.39 inches (This is a two-tailed test)
**2. Set the Significance Level**
* α = 0.05 (5% significance level)
**3. Calculate the Test Statistic**
* Since the population standard deviation is known, we will use a z-test.
* Given:
* Sample mean (x̄) = 67.47 inches
* Population mean (µ) = 67.39 inches
* Population standard deviation (σ) = 1.3 inches
* Sample size (n) = 400
* Formula for z-statistic:
* z = (x̄ - µ) / (σ / √n)
* Calculation:
* z = (67.47 - 67.39) / (1.3 / √400)
* z = 0.08 / (1.3 / 20)
* z = 0.08 / 0.065
* z ≈ 1.23
**4. Determine the Critical Values**
* For a two-tailed test at α = 0.05, the critical z-values are ±1.96.
**5. Make a Decision**
* Compare the calculated z-statistic (1.23) to the critical z-values (±1.96).
* Since -1.96 < 1.23 < 1.96, the calculated z-statistic falls within the acceptance region.
* Therefore, we fail to reject the null hypothesis.
**6. Draw a Conclusion**
* There is not enough evidence to reject the null hypothesis at the 5% significance level.
* We cannot reasonably say that the sample mean height is significantly different from the population mean height.
* In other words, it can be reasonably regarded as a sample from the population.
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