Intro to Business Statistics: Nonparametric Tests Sign Test Paired Data Single Sample Median

What This Is

The Sign Test is a non-parametric test used to compare the median of a single sample to a known population median or to compare the medians of two related samples. This test is useful when the data are paired or when the normality assumption is not met. For example, a retail chain wants to know if the average daily sales exceed $10,000. They collect data on daily sales for a week and want to compare the median sales to the known population median of $10,000.

Key Formulas & Symbols

• Sign Test Statistic (S): The number of positive differences between the paired observations.
• Test Statistic (V): The number of negative differences between the paired observations.
• Total Number of Signs (n): The total number of paired observations.
• Null Hypothesis (H₀): The median of the population is equal to the known population median.
• Alternative Hypothesis (H₁): The median of the population is not equal to the known population median.
• p-value: The probability of observing the test statistic (or more extreme) if the null hypothesis is true.
• Critical Value (CV): The value of the test statistic that separates the rejection region from the non-rejection region.
• Median (M): The middle value of the data when it is arranged in order.
• Wilcoxon Signed-Rank Test Statistic (T): The sum of the absolute values of the positive differences between the paired observations.

Step-by-Step Procedure

1. State Hypotheses: State the null and alternative hypotheses. For example, H₀: M = 10,000 and H₁: M ≠ 10,000.
2. Choose Test: Choose the Sign Test as the appropriate test for paired data or single sample median.
3. Compute Test Statistic: Compute the test statistic (S or V) by counting the number of positive or negative differences between the paired observations.
4. Find p-value or Critical Value: Find the p-value or critical value using a Sign Test table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Conclude whether to reject or fail to reject the null hypothesis based on the comparison.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the test statistic (or more extreme) if the null hypothesis is true. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to account for tied observations.
• Correction: Tied observations should be handled by assigning a rank to each tied observation and then summing the ranks.
• Mistake: Using the Sign Test when the data are not paired or when the normality assumption is met.
• Correction: The Sign Test is used for paired data or single sample median when the normality assumption is not met. Other tests, such as the t-test, should be used when the data are not paired or when the normality assumption is met.

Quick Practice Problems

1. A company wants to know if the median employee salary is greater than $50,000. They collect data on employee salaries and find that the median salary is $55,000. What is the p-value if the null hypothesis is that the median salary is equal to $50,000?

Final Answer: 0.02 Explanation: The p-value is the probability of observing a median salary of $55,000 (or more extreme) if the null hypothesis is true.

1. A retail chain wants to know if the median daily sales exceed $10,000. They collect data on daily sales for a week and find that the median sales is $12,000. What is the test statistic if the null hypothesis is that the median sales is equal to $10,000?

Final Answer: 2 Explanation: The test statistic is the number of positive differences between the paired observations, which is 2.

1. A company wants to know if the median customer satisfaction rating is greater than 4. They collect data on customer satisfaction ratings and find that the median rating is 4.5. What is the critical value if the significance level is 0.05?

Final Answer: 1.96 Explanation: The critical value is the value of the test statistic that separates the rejection region from the non-rejection region, which is 1.96.

Last-Minute Cram Sheet

• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• The Sign Test is used for paired data or single sample median when the normality assumption is not met.
• The test statistic (S or V) is the number of positive or negative differences between the paired observations.
• The p-value is found using a Sign Test table or calculator.
• The critical value is the value of the test statistic that separates the rejection region from the non-rejection region.
• The median (M) is the middle value of the data when it is arranged in order.
• The Wilcoxon Signed-Rank Test Statistic (T) is the sum of the absolute values of the positive differences between the paired observations.
• α = 0.05 is the default significance level.
• The Sign Test assumes that the data are paired and that the observations are independent.
• The Sign Test is a non-parametric test, meaning it does not assume a specific distribution of the data.