This article provides worked solutions to common problems encountered when testing statistical hypotheses. Understanding these examples is crucial for mastering the application of statistical methods. We'll cover various hypothesis tests, explaining the steps involved and the interpretation of the results.
Hypothesis Testing: A Quick Recap
Before diving into the worked solutions, let's briefly review the core components of hypothesis testing:
- Null Hypothesis (H₀): This is the statement we aim to disprove. It typically represents the status quo or a default assumption.
- Alternative Hypothesis (H₁ or Hₐ): This is the statement we are trying to prove. It contradicts the null hypothesis.
- Significance Level (α): The probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 and 0.01.
- Test Statistic: A calculated value based on sample data that is used to determine the p-value.
- P-value: The probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true.
- Decision Rule: If the p-value is less than or equal to the significance level (α), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Worked Solutions: Examples
Example 1: One-Sample t-test
Problem: A researcher claims that the average height of adult males in a certain city is 175 cm. A random sample of 100 adult males from this city yielded a mean height of 178 cm with a standard deviation of 5 cm. Test the researcher's claim at a 5% significance level.
Solution:
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State the hypotheses:
- H₀: μ = 175 cm (The average height is 175 cm)
- H₁: μ ≠ 175 cm (The average height is not 175 cm - two-tailed test)
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Determine the significance level: α = 0.05
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Calculate the test statistic: Using a one-sample t-test, we calculate: t = (178 - 175) / (5 / √100) = 6
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Determine the p-value: Using a t-distribution table with 99 degrees of freedom (df = n - 1), we find the p-value associated with t = 6. The p-value will be extremely small (much less than 0.05).
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Make a decision: Since the p-value is less than α, we reject the null hypothesis.
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Conclusion: There is sufficient evidence to reject the researcher's claim that the average height of adult males in the city is 175 cm.
Example 2: Two-Sample t-test (Independent Samples)
Problem: Two different teaching methods are compared. Group A (n=30) using method 1 has a mean test score of 75 with a standard deviation of 10. Group B (n=35) using method 2 has a mean test score of 80 with a standard deviation of 8. Test if there is a significant difference between the mean test scores of the two groups at a 1% significance level.
Solution: (Steps similar to Example 1, but using a two-sample t-test formula to calculate the test statistic and considering the appropriate degrees of freedom). The detailed calculation is omitted for brevity, but the key steps remain the same: defining hypotheses, determining significance level, calculating the test statistic, finding the p-value, making a decision based on the p-value and significance level and stating the conclusion.
Further Exploration
These are just two examples. Other common hypothesis tests include:
- Chi-square test: Used to analyze categorical data.
- ANOVA (Analysis of Variance): Used to compare means of three or more groups.
- Z-test: Used when the population standard deviation is known.
Mastering hypothesis testing requires practice. Work through various problems and understand the underlying principles to confidently analyze and interpret statistical results. Consult statistical textbooks and software for further assistance with more complex scenarios and calculations.
